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<p><strong>Description:</strong> An introduction to sums through examination of real–world problems like annuities. Covers finding closed form solutions and bounds with the perturbation, derivative, and integral methods.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="360">PROFESSOR: The</span> <span m="490">following</span> <span m="940">content</span> <span m="1530">is</span> <span m="1650">provided</span> <span m="2100">under</span> <span m="2350">a</span> <span m="2400">Creative</span> <span m="2800">Commons</span> <span m="3220">license.</span> <span m="4340">Your</span> <span m="4510">support</span> <span m="5050">will</span> <span m="5200">help</span> <span m="5500">MIT</span> <span m="5870">OpenCourseWare</span> <span m="6660">continue</span> <span m="7170">to</span> <span m="7290">offer</span> <span m="7690">high</span> <span m="7910">quality</span> <span m="8420">educational</span> <span m="9060">resources</span> <span m="9640">for</span> <span m="9820">free.</span> <span m="11020">To</span> <span m="11040">make</span> <span m="11250">a</span> <span m="11290">donation</span> <span m="11980">or</span> <span m="12230">view</span> <span m="12680">additional</span> <span m="13120">materials</span> <span m="13640">from</span> <span m="13830">hundreds</span> <span m="14240">of</span> <span m="14360">MIT</span> <span m="14760">courses,</span> <span m="15840">visit</span> <span m="16100">MIT</span> <span m="16510">OpenCourseWare</span> <span m="17560">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="22940">Today</span> <span m="23230">we're</span> <span m="23330">going</span> <span m="23450">to</span> <span m="23520">start</span> <span m="23820">with</span> <span m="23950">a</span> <span m="24000">simple</span> <span m="24350">problem</span> <span m="24740">that</span> <span m="24880">many</span> <span m="25220">of</span> <span m="25300">you</span> <span m="25800">may</span> <span m="25980">have</span> <span m="26110">already</span> <span m="26400">encountered.</span> <span m="27670">For</span> <span m="27800">example,</span> <span m="28220">when</span> <span m="28350">you</span> <span m="28410">got</span> <span m="28580">a</span> <span m="28620">student</span> <span m="28990">loan</span> <span m="29840">or</span> <span m="30240">if</span> <span m="30350">your</span> <span m="30620">family</span> <span m="31150">took</span> <span m="31350">a</span> <span m="31400">loan</span> <span m="31680">out</span> <span m="31880">on</span> <span m="32140">your</span> <span m="32299">house,</span> <span m="32564">you</span> <span m="32652">know,</span> <span m="32741">a</span> <span m="32830">home</span> <span m="33270">mortgage</span> <span m="33640">loan,</span> <span m="34540">and</span> <span m="34720">it's</span> <span m="34850">the</span> <span m="34940">problem</span> <span m="35310">of</span> <span m="35450">pricing</span> <span m="36070">an</span> <span m="36200">annuity.</span> <span m="37420">An</span> <span m="37480">annuity</span> <span m="37970">is</span> <span m="38150">a</span> <span m="38220">financial</span> <span m="38800">instrument</span> <span m="39720">that</span> <span m="40440">pays</span> <span m="40850">a</span> <span m="40900">fixed</span> <span m="41230">amount</span> <span m="41590">of</span> <span m="41690">money</span> <span m="42660">every</span> <span m="43080">year</span> <span m="43610">for</span> <span m="43790">some</span> <span m="44010">number</span> <span m="44320">of</span> <span m="44410">years,</span> <span m="44910">and</span> <span m="45050">it</span> <span m="45120">has</span> <span m="45330">a</span> <span m="45370">value</span> <span m="45830">associated</span> <span m="46400">with</span> <span m="46570">it.</span> </p>
<p><span m="47170">For</span> <span m="47200">example,</span> <span m="47720">with</span> <span m="47820">a</span> <span m="47920">student</span> <span m="48140">loan,</span> <span m="48390">the</span> <span m="48460">value</span> <span m="49150">is</span> <span m="49280">the</span> <span m="49340">amount</span> <span m="49570">of</span> <span m="49630">money</span> <span m="49950">they</span> <span m="50130">gave</span> <span m="50460">you</span> <span m="51190">to</span> <span m="51310">pay</span> <span m="51540">MIT,</span> <span m="52540">then</span> <span m="52750">later</span> <span m="53020">in</span> <span m="53100">life--</span> <span m="54080">every</span> <span m="54550">year</span> <span m="55160">or</span> <span m="55350">every</span> <span m="55680">month--</span> <span m="56070">you're</span> <span m="56180">going</span> <span m="56300">to</span> <span m="56370">send</span> <span m="56550">them</span> <span m="56630">a</span> <span m="56690">check.</span> <span m="57860">And</span> <span m="57950">you</span> <span m="58030">want</span> <span m="58220">to</span> <span m="58240">sort</span> <span m="58260">of</span> <span m="58280">equate</span> <span m="58890">those</span> <span m="59130">two</span> <span m="59300">things</span> <span m="59690">to</span> <span m="59770">find</span> <span m="60100">out,</span> <span m="60960">are</span> <span m="61310">you</span> <span m="61410">getting</span> <span m="61640">enough</span> <span m="61860">money</span> <span m="62150">for</span> <span m="62270">the</span> <span m="62350">money</span> <span m="62570">you're</span> <span m="62660">going</span> <span m="62780">to</span> <span m="62840">pay</span> <span m="63030">back</span> <span m="63290">monthly</span> <span m="63680">sometime</span> <span m="64150">in</span> <span m="64220">the</span> <span m="64280">future?</span> </p>
<p><span m="65489">So</span> <span m="65650">let's</span> <span m="65860">define</span> <span m="66320">this,</span> <span m="68310">and</span> <span m="68420">there's</span> <span m="68560">a</span> <span m="68620">lot</span> <span m="68920">of</span> <span m="68990">variations</span> <span m="69990">on</span> <span m="70230">annuities,</span> <span m="71970">but</span> <span m="72230">we'll</span> <span m="72490">start</span> <span m="72830">with</span> <span m="73010">one</span> <span m="73360">that's</span> <span m="73520">called</span> <span m="74430">an</span> <span m="74620">n-year</span> <span m="78580">$m-dollar</span> <span m="81170">payment</span> <span m="83690">annuity,</span> <span m="90450">and</span> <span m="90640">it</span> <span m="90750">works</span> <span m="91090">by</span> <span m="91290">paying</span> <span m="93190">m</span> <span m="93530">dollars</span> <span m="95890">at</span> <span m="96080">the</span> <span m="96160">start</span> <span m="96610">of</span> <span m="96700">each</span> <span m="96910">year,</span> <span m="106950">and</span> <span m="107130">it</span> <span m="107230">lasts</span> <span m="107670">for</span> <span m="107830">n</span> <span m="108120">years.</span> <span m="112770">Now,</span> <span m="112960">usually</span> <span m="113640">n</span> <span m="113840">is</span> <span m="114000">finite,</span> <span m="114480">but</span> <span m="114630">not</span> <span m="114860">always,</span> <span m="115380">and</span> <span m="115620">in</span> <span m="115800">a</span> <span m="115840">few</span> <span m="116010">minutes</span> <span m="116390">we'll</span> <span m="116490">talk</span> <span m="116610">about</span> <span m="116770">the</span> <span m="116840">case</span> <span m="117160">when</span> <span m="117280">it's</span> <span m="117430">infinite.</span> </p>
<p><span m="119150">But</span> <span m="119320">this</span> <span m="119520">includes</span> <span m="120930">home</span> <span m="121210">mortgages,</span> <span m="121850">where</span> <span m="121960">you</span> <span m="122060">pay</span> <span m="122320">every</span> <span m="122500">month</span> <span m="122710">for</span> <span m="122810">30</span> <span m="123110">years.</span> <span m="123690">Megabucks,</span> <span m="124420">the</span> <span m="124510">lottery.</span> <span m="125800">They</span> <span m="125940">don't</span> <span m="126080">actually</span> <span m="126370">give</span> <span m="126560">you</span> <span m="126640">the</span> <span m="126720">million</span> <span m="127040">dollars.</span> <span m="127890">They</span> <span m="128039">give</span> <span m="128240">you</span> <span m="128340">$50,000</span> <span m="129610">per</span> <span m="129850">year</span> <span m="130120">for</span> <span m="130250">20</span> <span m="130570">years,</span> <span m="131030">and</span> <span m="131170">call</span> <span m="131430">it</span> <span m="131490">a</span> <span m="131530">million</span> <span m="131840">bucks.</span> <span m="133250">Retirement</span> <span m="133900">plans.</span> <span m="134880">You</span> <span m="135150">pay</span> <span m="135550">in</span> <span m="136250">every</span> <span m="136610">year,</span> <span m="136940">and</span> <span m="137050">then</span> <span m="137190">you</span> <span m="137270">get</span> <span m="137840">some</span> <span m="138640">big</span> <span m="138890">lump</span> <span m="139090">sum</span> <span m="139310">later.</span> <span m="139620">Life</span> <span m="139920">insurance</span> <span m="140500">benefits,</span> <span m="141920">you</span> <span m="142020">know,</span> <span m="142140">and</span> <span m="142230">so</span> <span m="142410">forth.</span> </p>
<p><span m="142880">Now,</span> <span m="143850">if</span> <span m="144030">you</span> <span m="144110">go</span> <span m="144210">to</span> <span m="144310">Wall</span> <span m="144600">Street,</span> <span m="144860">this</span> <span m="145010">is</span> <span m="145100">a</span> <span m="145150">big</span> <span m="145390">deal.</span> <span m="145940">A</span> <span m="145990">lot</span> <span m="146440">of</span> <span m="146500">the</span> <span m="146570">stuff</span> <span m="146820">that</span> <span m="146930">happens</span> <span m="147290">on</span> <span m="147420">Wall</span> <span m="147700">Street</span> <span m="148080">involves</span> <span m="148760">annuities</span> <span m="149730">in</span> <span m="149860">one</span> <span m="150100">form</span> <span m="150320">or</span> <span m="150350">another,</span> <span m="150580">packaging</span> <span m="151170">them</span> <span m="151310">all</span> <span m="151570">up</span> <span m="152400">and,</span> <span m="152890">in</span> <span m="152980">fact,</span> <span m="153730">we'll</span> <span m="154230">look</span> <span m="154430">at</span> <span m="154560">later</span> <span m="154800">when</span> <span m="154950">we</span> <span m="154990">do</span> <span m="155030">probability.</span> <span m="156120">It</span> <span m="156370">was</span> <span m="156540">how</span> <span m="156780">these</span> <span m="157020">things</span> <span m="157230">were</span> <span m="157330">packaged</span> <span m="157860">and</span> <span m="157970">sold</span> <span m="158380">that</span> <span m="158510">led</span> <span m="158710">to</span> <span m="158780">the</span> <span m="159290">sub-prime</span> <span m="159850">mortgage</span> <span m="160220">disaster,</span> <span m="161270">and</span> <span m="161410">we'll</span> <span m="161500">see</span> <span m="161720">how</span> <span m="161890">some</span> <span m="162070">confusion</span> <span m="162650">over</span> <span m="163350">independents,</span> <span m="164160">when</span> <span m="164270">you</span> <span m="164370">look</span> <span m="164560">at</span> <span m="164980">random</span> <span m="165360">variables,</span> <span m="165940">led</span> <span m="166190">to</span> <span m="166700">the</span> <span m="166810">global</span> <span m="167150">recession,</span> <span m="167680">a</span> <span m="167790">real</span> <span m="167960">disaster.</span> </p>
<p><span m="169040">Of</span> <span m="169100">course,</span> <span m="169300">some</span> <span m="169520">people</span> <span m="169880">understood</span> <span m="170380">how</span> <span m="170510">all</span> <span m="170660">that</span> <span m="170850">worked,</span> <span m="171220">made</span> <span m="171520">hundreds</span> <span m="172120">of</span> <span m="172190">billions</span> <span m="172490">of</span> <span m="172550">dollars</span> <span m="173090">at</span> <span m="173150">the</span> <span m="173220">same</span> <span m="173500">time.</span> <span m="173820">So</span> <span m="173990">it</span> <span m="174100">was</span> <span m="174220">sort</span> <span m="174370">of</span> <span m="175030">money</span> <span m="175270">went</span> <span m="175450">from</span> <span m="175570">one</span> <span m="175730">place</span> <span m="175970">to</span> <span m="176090">another.</span> <span m="177260">So</span> <span m="177360">it's</span> <span m="177730">pretty</span> <span m="177940">important</span> <span m="178380">to</span> <span m="178440">understand</span> <span m="179790">how</span> <span m="180060">much</span> <span m="180570">this</span> <span m="180940">is</span> <span m="181140">worth.</span> <span m="182930">What</span> <span m="183250">is</span> <span m="183910">this</span> <span m="184180">instrument--</span> <span m="184790">a</span> <span m="184870">piece</span> <span m="185120">of</span> <span m="185200">paper</span> <span m="185550">that</span> <span m="185740">says</span> <span m="186010">it</span> <span m="186080">will</span> <span m="186170">pay</span> <span m="186470">you</span> <span m="186620">m</span> <span m="186770">dollars</span> <span m="187290">at</span> <span m="187370">the</span> <span m="187430">beginning</span> <span m="187710">of</span> <span m="187780">each</span> <span m="187940">year</span> <span m="188100">for</span> <span m="188230">n</span> <span m="188370">years,</span> <span m="188900">what</span> <span m="189160">is</span> <span m="189300">that</span> <span m="189530">worth</span> <span m="190290">today?</span> </p>
<p><span m="191430">For</span> <span m="191580">example,</span> <span m="193130">say</span> <span m="193400">I</span> <span m="193480">gave</span> <span m="193740">you</span> <span m="193880">a</span> <span m="193930">choice--</span> <span m="194650">the</span> <span m="194730">Megabucks</span> <span m="195260">choice--</span> <span m="196310">$50,000</span> <span m="197200">a</span> <span m="197250">year</span> <span m="197580">for</span> <span m="197710">20</span> <span m="198070">years</span> <span m="199320">or</span> <span m="199510">a</span> <span m="199560">million</span> <span m="199910">dollars</span> <span m="200630">today.</span> <span m="201950">How</span> <span m="202140">many</span> <span m="202330">people</span> <span m="202640">would</span> <span m="202770">prefer</span> <span m="203200">the</span> <span m="203290">$50,000</span> <span m="204050">a</span> <span m="204090">year</span> <span m="204270">for</span> <span m="204400">20</span> <span m="204700">years?</span> <span m="206430">A</span> <span m="206530">few</span> <span m="206740">of</span> <span m="206885">you.</span> <span m="207030">How</span> <span m="207190">many</span> <span m="207350">would</span> <span m="207460">prefer</span> <span m="207760">the</span> <span m="207850">million</span> <span m="208340">bucks</span> <span m="208830">right</span> <span m="209060">up</span> <span m="209170">front?</span> <span m="210840">Much</span> <span m="211120">better.</span> <span m="211670">OK.</span> <span m="212630">Always</span> <span m="213160">better</span> <span m="213380">to</span> <span m="213480">have</span> <span m="213680">the</span> <span m="213760">cash</span> <span m="214130">in</span> <span m="214260">hand,</span> <span m="214970">because</span> <span m="215310">there's</span> <span m="215490">things</span> <span m="215710">like</span> <span m="215890">inflation--</span> <span m="216900">pretty</span> <span m="217130">low</span> <span m="217370">now--</span> <span m="218820">interest.</span> <span m="219680">You</span> <span m="219960">can</span> <span m="220060">put</span> <span m="220230">the</span> <span m="220290">money</span> <span m="220580">in</span> <span m="220660">the</span> <span m="220740">bank</span> <span m="221160">or</span> <span m="221240">invest</span> <span m="221650">it</span> <span m="221740">and</span> <span m="221880">make</span> <span m="222110">some</span> <span m="222280">money</span> <span m="222840">hopefully.</span> </p>
<p><span m="224980">So</span> <span m="225180">the</span> <span m="225280">million</span> <span m="225630">dollars</span> <span m="225960">today</span> <span m="226180">is</span> <span m="226260">a</span> <span m="226320">lot</span> <span m="226540">better,</span> <span m="227170">which</span> <span m="227430">is</span> <span m="227530">why</span> <span m="227720">the</span> <span m="227810">State</span> <span m="228240">pays</span> <span m="228530">you</span> <span m="229312">50</span> <span m="229660">grand</span> <span m="229940">a</span> <span m="230000">year</span> <span m="230150">for</span> <span m="230270">20</span> <span m="230540">years.</span> <span m="230960">It's</span> <span m="231240">better</span> <span m="231470">for</span> <span m="231610">them,</span> <span m="231850">and</span> <span m="231950">they</span> <span m="232000">call</span> <span m="232280">it</span> <span m="232340">a</span> <span m="232370">million</span> <span m="232670">bucks.</span> <span m="233810">So</span> <span m="233960">that</span> <span m="234090">was</span> <span m="234240">pretty</span> <span m="234470">clear,</span> <span m="234710">but</span> <span m="234840">what</span> <span m="235030">if</span> <span m="235130">I</span> <span m="235220">gave</span> <span m="235450">you</span> <span m="235550">this</span> <span m="235840">option--</span> <span m="237720">700</span> <span m="238580">grand</span> <span m="238950">today</span> <span m="240400">or</span> <span m="240830">50</span> <span m="241330">grand</span> <span m="241680">a</span> <span m="241750">year</span> <span m="241950">for</span> <span m="242070">20</span> <span m="242420">years?</span> <span m="243700">How</span> <span m="243860">many</span> <span m="244050">people</span> <span m="244400">want</span> <span m="244630">the</span> <span m="244710">cash</span> <span m="245100">upfront--</span> <span m="245510">700</span> <span m="246170">grand</span> <span m="246560">only.</span> <span m="248090">A</span> <span m="248170">few.</span> </p>
<p><span m="248660">How</span> <span m="248820">many</span> <span m="249010">people</span> <span m="249300">want</span> <span m="250410">50</span> <span m="250720">grand</span> <span m="250970">a</span> <span m="251030">year</span> <span m="251190">for</span> <span m="251300">20</span> <span m="251560">years?</span> <span m="253310">All</span> <span m="253400">right,</span> <span m="253440">we're</span> <span m="253580">almost--</span> <span m="253990">that's</span> <span m="254170">pretty</span> <span m="254360">close</span> <span m="254600">to</span> <span m="254700">half</span> <span m="254900">way.</span> <span m="255770">How</span> <span m="256029">about</span> <span m="256420">500</span> <span m="257550">grand</span> <span m="258019">today</span> <span m="258529">versus</span> <span m="259000">50</span> <span m="259430">grand</span> <span m="259800">a</span> <span m="259850">year</span> <span m="260000">for</span> <span m="260120">20</span> <span m="260420">years?</span> <span m="260750">How</span> <span m="260899">many</span> <span m="261060">want</span> <span m="261930">half</span> <span m="262200">a</span> <span m="262250">million</span> <span m="262620">today?</span> <span m="264670">A</span> <span m="264770">lot</span> <span m="264890">of</span> <span m="264950">people</span> <span m="265130">like</span> <span m="265310">the</span> <span m="265390">cash.</span> <span m="265760">You</span> <span m="265820">know,</span> <span m="265910">it's</span> <span m="266030">this</span> <span m="266190">kind</span> <span m="266360">of</span> <span m="266430">a</span> <span m="266470">time,</span> <span m="266830">you</span> <span m="266900">have</span> <span m="267020">the</span> <span m="267090">recession,</span> <span m="268140">it's</span> <span m="268240">a</span> <span m="268290">disaster</span> <span m="268860">on</span> <span m="268980">Wall</span> <span m="269360">Street.</span> <span m="269740">You</span> <span m="269930">know,</span> <span m="270120">Street</span> <span m="270270">Wall</span> <span m="270460">Street</span> <span m="270610">didn't</span> <span m="270740">like</span> <span m="270900">the</span> <span m="270970">cash.</span> </p>
<p><span m="271700">How</span> <span m="271860">many</span> <span m="272050">want,</span> <span m="272630">instead</span> <span m="273050">of</span> <span m="273130">a</span> <span m="273190">half</span> <span m="273450">a</span> <span m="273670">million</span> <span m="274100">up</span> <span m="274260">front,</span> <span m="274730">50</span> <span m="275040">grand</span> <span m="275300">a</span> <span m="275330">year</span> <span m="275430">for</span> <span m="275530">20</span> <span m="275790">years?</span> <span m="277260">All</span> <span m="277340">right,</span> <span m="277420">now</span> <span m="277670">we're</span> <span m="277730">about</span> <span m="277990">halfway.</span> <span m="278710">All</span> <span m="278890">right,</span> <span m="279040">well</span> <span m="279190">that's</span> <span m="279420">pretty</span> <span m="279630">good.</span> <span m="279850">So</span> <span m="279910">we're</span> <span m="279970">going</span> <span m="280090">to</span> <span m="280150">find</span> <span m="280430">out</span> <span m="280550">what</span> <span m="280690">you</span> <span m="280770">should</span> <span m="281000">pay,</span> <span m="281815">or</span> <span m="282290">at</span> <span m="282350">least</span> <span m="282550">one</span> <span m="282740">way</span> <span m="282950">of</span> <span m="283300">estimating</span> <span m="283820">that.</span> </p>
<p><span m="284750">Now,</span> <span m="284920">to</span> <span m="285020">do</span> <span m="285240">that,</span> <span m="286520">we've</span> <span m="286690">got</span> <span m="286800">to</span> <span m="286860">figure</span> <span m="287210">out</span> <span m="287500">what</span> <span m="288460">$1</span> <span m="289370">today</span> <span m="290630">is</span> <span m="290820">worth</span> <span m="291130">in</span> <span m="291210">a</span> <span m="291300">year.</span> <span m="293670">And</span> <span m="293920">to</span> <span m="293990">do</span> <span m="294210">that,</span> <span m="294860">we</span> <span m="295050">make</span> <span m="295300">an</span> <span m="295380">assumption,</span> <span m="300680">and</span> <span m="300750">the</span> <span m="300820">assumption</span> <span m="301300">is</span> <span m="301510">that</span> <span m="301610">there's</span> <span m="301780">a</span> <span m="301840">fixed--</span> <span m="303750">we'll</span> <span m="303890">call</span> <span m="304140">it</span> <span m="304200">an</span> <span m="304350">interest</span> <span m="304800">rate.</span> <span m="306600">It's</span> <span m="306810">sort</span> <span m="306990">of</span> <span m="307060">the</span> <span m="307420">devaluation</span> <span m="308290">of</span> <span m="308350">the</span> <span m="308420">money</span> <span m="308940">per</span> <span m="309120">year.</span> <span m="312180">And</span> <span m="312340">we're</span> <span m="312470">going</span> <span m="312590">to</span> <span m="312650">call</span> <span m="312930">it</span> <span m="313040">p.</span> <span m="313490">Later</span> <span m="313760">we'll</span> <span m="313890">plug</span> <span m="314100">in</span> <span m="314310">values</span> <span m="314690">for</span> <span m="314800">p,</span> <span m="315000">but</span> <span m="315120">you</span> <span m="315240">can</span> <span m="315360">think</span> <span m="315570">of</span> <span m="315650">it</span> <span m="315740">as</span> <span m="315880">like</span> <span m="316600">6%,</span> <span m="317900">1%.</span> <span m="318890">You</span> <span m="319010">know,</span> <span m="319150">it's</span> <span m="319290">the</span> <span m="319520">money,</span> <span m="319840">if</span> <span m="319930">you</span> <span m="320010">put</span> <span m="320210">money</span> <span m="320400">in</span> <span m="320480">a</span> <span m="320530">bank,</span> <span m="320880">they'll</span> <span m="321000">give</span> <span m="321180">you</span> <span m="321280">some</span> <span m="321540">percent</span> <span m="321940">back</span> <span m="322300">every</span> <span m="322500">year.</span> <span m="323650">And,</span> <span m="323780">of</span> <span m="323850">course,</span> <span m="324640">the</span> <span m="324740">fact</span> <span m="325040">that</span> <span m="325230">different</span> <span m="325600">people</span> <span m="325830">have</span> <span m="325940">different</span> <span m="326230">ideas</span> <span m="326590">of</span> <span m="326690">what</span> <span m="326860">this</span> <span m="327020">would</span> <span m="327150">be,</span> <span m="327430">allows</span> <span m="327790">people</span> <span m="328020">to</span> <span m="328080">make</span> <span m="328260">money</span> <span m="328510">on</span> <span m="328620">Wall</span> <span m="328860">Street.</span> <span m="329950">As</span> <span m="330160">we'll</span> <span m="330290">see,</span> <span m="330590">a</span> <span m="330810">slight</span> <span m="331290">difference</span> <span m="331640">in</span> <span m="332090">p</span> <span m="332440">can</span> <span m="332615">make</span> <span m="333130">big</span> <span m="333330">differences</span> <span m="333820">in</span> <span m="333920">what</span> <span m="334110">the</span> <span m="334610">annuity</span> <span m="335020">is</span> <span m="335150">worth.</span> </p>
<p><span m="336460">So</span> <span m="336700">for</span> <span m="336830">example,</span> <span m="339750">$1</span> <span m="340380">today</span> <span m="345580">is</span> <span m="345740">going</span> <span m="345950">to</span> <span m="346020">equal</span> <span m="347460">1</span> <span m="347850">plus</span> <span m="348130">p</span> <span m="348380">dollars</span> <span m="349830">in</span> <span m="349980">one</span> <span m="350200">year.</span> <span m="354360">Similarly,</span> <span m="354960">$1</span> <span m="355490">today--</span> <span m="356670">how</span> <span m="356850">much</span> <span m="357040">is</span> <span m="357130">that</span> <span m="357290">going</span> <span m="357410">to</span> <span m="357470">be</span> <span m="357540">worth</span> <span m="358760">in</span> <span m="358960">two</span> <span m="359150">years?</span> <span m="363910">Say</span> <span m="364160">that</span> <span m="364250">p</span> <span m="364520">stays</span> <span m="364800">fixed,</span> <span m="365270">the</span> <span m="365400">same</span> <span m="365720">over</span> <span m="365900">all</span> <span m="366080">time.</span> <span m="370060">One</span> <span m="370390">plus</span> <span m="370680">p</span> <span m="370970">squared,</span> <span m="373410">because</span> <span m="373770">every</span> <span m="374030">year</span> <span m="374240">you</span> <span m="374360">multiply</span> <span m="374870">what</span> <span m="375060">you</span> <span m="375150">got</span> <span m="375470">by</span> <span m="375570">1</span> <span m="375840">plus</span> <span m="376100">p,</span> <span m="376570">because</span> <span m="376750">that's</span> <span m="376970">the</span> <span m="377100">interest</span> <span m="377410">you're</span> <span m="377500">getting.</span> <span m="378560">All</span> <span m="378625">right,</span> <span m="378690">we'll</span> <span m="378810">think</span> <span m="379020">of</span> <span m="379090">it</span> <span m="379150">in</span> <span m="379240">terms</span> <span m="379520">of</span> <span m="379610">interest.</span> <span m="380940">In</span> <span m="381090">three</span> <span m="381460">years,</span> <span m="382050">$1</span> <span m="382510">today</span> <span m="383510">is</span> <span m="383750">worth</span> <span m="385250">1</span> <span m="385490">plus</span> <span m="385760">p</span> <span m="385950">cubed</span> <span m="387900">in</span> <span m="388070">three</span> <span m="388330">years</span> <span m="389380">and</span> <span m="389590">so</span> <span m="389790">forth.</span> </p>
<p><span m="392740">All</span> <span m="392920">right.</span> <span m="393160">Now,</span> <span m="393650">what</span> <span m="393760">we</span> <span m="393880">really</span> <span m="394230">care</span> <span m="394540">about</span> <span m="394850">is</span> <span m="394970">what's</span> <span m="395190">$1,</span> <span m="395870">or</span> <span m="396030">m</span> <span m="396300">dollars,</span> <span m="396750">worth</span> <span m="397610">today</span> <span m="398360">if</span> <span m="398460">you're</span> <span m="398580">getting</span> <span m="398890">it</span> <span m="399220">next</span> <span m="399570">year?</span> <span m="400090">So</span> <span m="400120">we</span> <span m="400190">need</span> <span m="400330">to</span> <span m="400410">sort</span> <span m="400640">of</span> <span m="401110">flip</span> <span m="401610">this</span> <span m="403000">back</span> <span m="403310">the</span> <span m="403430">other</span> <span m="403580">way.</span> <span m="407930">So</span> <span m="408390">what</span> <span m="408670">is</span> <span m="409300">$1</span> <span m="411210">in</span> <span m="411370">a</span> <span m="411470">year</span> <span m="412230">worth</span> <span m="412590">today</span> <span m="415700">in</span> <span m="415780">terms</span> <span m="416050">of</span> <span m="416140">p?</span> <span m="419160">So</span> <span m="419300">if</span> <span m="419380">I'm</span> <span m="419510">going</span> <span m="419630">to</span> <span m="419690">be</span> <span m="419750">paid</span> <span m="420030">$1</span> <span m="420470">in</span> <span m="420560">a</span> <span m="420610">year,</span> <span m="421870">what</span> <span m="422040">would</span> <span m="422160">be</span> <span m="422290">the</span> <span m="422380">equivalent</span> <span m="422920">amount</span> <span m="423170">to</span> <span m="423230">be</span> <span m="423330">paid</span> <span m="423650">today?</span> <span m="427580">One</span> <span m="428040">over</span> <span m="428240">1</span> <span m="428490">plus</span> <span m="428770">p,</span> <span m="431680">because</span> <span m="432020">what's</span> <span m="432190">happening</span> <span m="432640">here</span> <span m="433480">is,</span> <span m="434010">as</span> <span m="434150">you</span> <span m="434230">go</span> <span m="434390">forward</span> <span m="434880">in</span> <span m="434980">a</span> <span m="435080">year,</span> <span m="435420">you</span> <span m="435500">just</span> <span m="435700">multiply</span> <span m="436120">by</span> <span m="436260">1</span> <span m="436550">plus</span> <span m="436790">p.</span> <span m="437690">So</span> <span m="437930">1</span> <span m="438530">over</span> <span m="438870">1</span> <span m="439110">plus</span> <span m="439380">p</span> <span m="439890">turns</span> <span m="440240">into</span> <span m="440820">$1</span> <span m="442890">in</span> <span m="443040">a</span> <span m="443110">year--</span> <span m="444270">being</span> <span m="444510">paid</span> <span m="444750">in</span> <span m="444810">a</span> <span m="444900">year.</span> <span m="447270">All</span> <span m="447500">right.</span> <span m="448130">What</span> <span m="448490">is</span> <span m="449270">$1</span> <span m="449960">a</span> <span m="450090">year</span> <span m="450730">in</span> <span m="450890">two</span> <span m="451170">years</span> <span m="454250">worth</span> <span m="454590">today?</span> <span m="457650">One</span> <span m="458370">over</span> <span m="458760">1</span> <span m="459060">plus</span> <span m="459380">p</span> <span m="460270">squared.</span> <span m="462610">So</span> <span m="462790">$1</span> <span m="463195">in</span> <span m="463600">two</span> <span m="463775">years</span> <span m="463950">is</span> <span m="464070">worth</span> <span m="464460">this</span> <span m="464710">much</span> <span m="465250">today.</span> </p>
<p><span m="468060">Well,</span> <span m="468380">now</span> <span m="468770">we</span> <span m="468910">can</span> <span m="469140">use</span> <span m="469530">this</span> <span m="470070">to</span> <span m="470200">go</span> <span m="470370">figure</span> <span m="470800">out</span> <span m="471880">the</span> <span m="472000">current</span> <span m="472530">value</span> <span m="473040">of</span> <span m="473140">that</span> <span m="473350">annuity.</span> <span m="475080">We</span> <span m="475210">just</span> <span m="475410">figure</span> <span m="475660">out</span> <span m="475800">what</span> <span m="475930">every</span> <span m="476170">payment</span> <span m="476600">is</span> <span m="476700">worth</span> <span m="478130">today</span> <span m="478930">and</span> <span m="479030">then</span> <span m="479060">add</span> <span m="479230">it</span> <span m="479320">up.</span> <span m="481380">So</span> <span m="481530">we'll</span> <span m="481650">put</span> <span m="481830">the</span> <span m="481910">payments</span> <span m="482580">over</span> <span m="482800">here,</span> <span m="484220">and</span> <span m="484380">we'll</span> <span m="484470">compute</span> <span m="484880">the</span> <span m="484940">current</span> <span m="485380">value</span> <span m="486890">of</span> <span m="487070">every</span> <span m="487280">payment</span> <span m="487880">on</span> <span m="488050">this</span> <span m="488200">side.</span> <span m="491840">So</span> <span m="492080">with</span> <span m="492190">the</span> <span m="492270">annuity,</span> <span m="493150">the</span> <span m="493260">way</span> <span m="493390">we've</span> <span m="493580">set</span> <span m="493840">it</span> <span m="493930">up</span> <span m="494150">is</span> <span m="494250">it</span> <span m="494360">pays</span> <span m="494800">n</span> <span m="495060">dollars</span> <span m="495460">at</span> <span m="495530">the</span> <span m="495610">start</span> <span m="495980">of</span> <span m="496080">every</span> <span m="496290">year,</span> <span m="496630">so</span> <span m="496750">the</span> <span m="496820">first</span> <span m="497670">payment</span> <span m="498040">is</span> <span m="498170">now.</span> <span m="499970">So</span> <span m="500150">the</span> <span m="500240">first</span> <span m="500610">of</span> <span m="500680">the</span> <span m="500790">n</span> <span m="500900">payments</span> <span m="502080">is</span> <span m="502390">now,</span> <span m="503710">and</span> <span m="504100">since</span> <span m="504300">it's</span> <span m="504420">being</span> <span m="504670">paid</span> <span m="505010">now,</span> <span m="505380">that's</span> <span m="505620">worth</span> <span m="505940">m</span> <span m="506290">dollars.</span> <span m="507130">There's</span> <span m="507350">no</span> <span m="508540">devaluation.</span> </p>
<p><span m="510320">The</span> <span m="510430">next</span> <span m="510750">payment</span> <span m="511430">is</span> <span m="511810">m</span> <span m="512049">dollars</span> <span m="514280">in</span> <span m="514490">one</span> <span m="514750">year,</span> <span m="516789">and</span> <span m="517010">so</span> <span m="517120">that's</span> <span m="517400">going</span> <span m="517530">to</span> <span m="517590">be</span> <span m="517700">worth</span> <span m="519730">m</span> <span m="520230">over</span> <span m="520460">1</span> <span m="520679">plus</span> <span m="520970">p</span> <span m="521169">today.</span> <span m="523360">And</span> <span m="523510">the</span> <span m="523570">next</span> <span m="523850">payment</span> <span m="524420">is</span> <span m="525110">m</span> <span m="525360">dollars</span> <span m="525930">in</span> <span m="526160">two</span> <span m="526420">years.</span> <span m="528850">That's</span> <span m="529250">worth</span> <span m="530760">m</span> <span m="531150">over</span> <span m="531690">1</span> <span m="531930">plus</span> <span m="532220">p</span> <span m="532400">squared,</span> <span m="533870">and</span> <span m="534010">we</span> <span m="534100">keep</span> <span m="534390">on</span> <span m="534600">going</span> <span m="535480">until</span> <span m="535770">the</span> <span m="535840">last</span> <span m="536340">payment.</span> <span m="537880">It's</span> <span m="538130">the</span> <span m="538310">n-th</span> <span m="538820">payment,</span> <span m="539570">so</span> <span m="539750">it's</span> <span m="539920">m</span> <span m="540160">dollars</span> <span m="540790">in</span> <span m="541940">n</span> <span m="542200">minus</span> <span m="542530">1</span> <span m="543200">years.</span> <span m="546330">And</span> <span m="546540">so</span> <span m="547410">that's</span> <span m="547770">going</span> <span m="547890">to</span> <span m="547970">be</span> <span m="548140">worth</span> <span m="548630">m</span> <span m="550740">over</span> <span m="551220">1</span> <span m="551560">plus</span> <span m="551880">p</span> <span m="553260">to</span> <span m="553380">the</span> <span m="553510">n</span> <span m="553650">minus</span> <span m="553970">1.</span> </p>
<p><span m="557120">All</span> <span m="557230">right.</span> <span m="557390">So</span> <span m="557480">we</span> <span m="557560">can</span> <span m="557690">compute</span> <span m="558060">the</span> <span m="558130">current</span> <span m="558540">value</span> <span m="558860">of</span> <span m="558950">all</span> <span m="559160">those</span> <span m="559380">payments,</span> <span m="560440">then</span> <span m="560610">the</span> <span m="560740">annuity</span> <span m="562620">is</span> <span m="562800">computed--</span> <span m="563330">the</span> <span m="563390">value</span> <span m="563750">is</span> <span m="563840">computed</span> <span m="564250">just</span> <span m="564430">by</span> <span m="564630">adding</span> <span m="564990">these</span> <span m="565210">up,</span> <span m="566780">of</span> <span m="566890">all</span> <span m="567060">the</span> <span m="567130">current</span> <span m="567440">values.</span> <span m="568820">So</span> <span m="568970">the</span> <span m="569050">total</span> <span m="569410">current</span> <span m="569720">value</span> <span m="573040">is</span> <span m="573440">the</span> <span m="573530">sum</span> <span m="575860">i</span> <span m="576150">equals</span> <span m="576590">0</span> <span m="577070">to</span> <span m="577180">n</span> <span m="577290">minus</span> <span m="577650">1</span> <span m="579012">of</span> <span m="579498">m</span> <span m="580470">over</span> <span m="580890">1</span> <span m="581250">plus</span> <span m="581580">p</span> <span m="582650">to</span> <span m="582780">the</span> <span m="582970">i.</span> <span m="584490">And</span> <span m="584620">that</span> <span m="584810">is</span> <span m="584960">the</span> <span m="585070">total</span> <span m="585510">current</span> <span m="585840">value.</span> <span m="590300">That's</span> <span m="590630">what</span> <span m="590800">you</span> <span m="590910">should</span> <span m="591130">pay</span> <span m="591440">today</span> <span m="594670">for</span> <span m="594940">the</span> <span m="595050">annuity.</span> <span m="596830">Any</span> <span m="597010">questions?</span> <span m="599590">What</span> <span m="599730">we</span> <span m="599820">did</span> <span m="600020">here?</span> </p>
<p><span m="602150">All</span> <span m="602400">right.</span> <span m="603890">Well,</span> <span m="605040">of</span> <span m="605100">course,</span> <span m="605370">what</span> <span m="605470">we'd</span> <span m="605500">like</span> <span m="606050">is</span> <span m="606190">a</span> <span m="606240">closed</span> <span m="606640">form</span> <span m="606880">expression</span> <span m="607370">here.</span> <span m="608530">Something</span> <span m="608870">that's</span> <span m="608990">simple</span> <span m="609610">so</span> <span m="609740">we</span> <span m="609820">could</span> <span m="609930">actually</span> <span m="610260">get</span> <span m="610430">a</span> <span m="610470">feel</span> <span m="610760">without</span> <span m="610990">having</span> <span m="611160">to</span> <span m="611560">add</span> <span m="611810">up</span> <span m="611930">all</span> <span m="612120">those</span> <span m="612340">terms,</span> <span m="614980">and</span> <span m="615470">that's</span> <span m="615650">not</span> <span m="615810">hard</span> <span m="615990">to</span> <span m="616050">get.</span> <span m="616520">In</span> <span m="616650">fact,</span> <span m="617880">let's</span> <span m="619030">put</span> <span m="619350">this</span> <span m="619790">sum</span> <span m="620070">in</span> <span m="620150">a</span> <span m="620190">form</span> <span m="620560">that</span> <span m="620680">might</span> <span m="621000">be</span> <span m="621120">more</span> <span m="621320">familiar.</span> </p>
<p><span m="623420">This</span> <span m="623580">equals--</span> <span m="624280">we'll</span> <span m="624440">pull</span> <span m="624670">the</span> <span m="624820">m</span> <span m="625030">out</span> <span m="625750">in</span> <span m="625890">front--</span> <span m="627310">and</span> <span m="627530">let's</span> <span m="627890">use</span> <span m="628400">x</span> <span m="628710">to</span> <span m="628800">be</span> <span m="628890">1</span> <span m="629160">over</span> <span m="629320">1</span> <span m="629500">plus</span> <span m="629710">p</span> <span m="629820">to</span> <span m="629910">the</span> <span m="630040">i.</span> <span m="637140">And</span> <span m="637270">so</span> <span m="637730">x</span> <span m="640740">equals</span> <span m="641220">1</span> <span m="641440">over</span> <span m="641570">1</span> <span m="641740">plus</span> <span m="641980">p,</span> <span m="646000">and</span> <span m="646140">I</span> <span m="646360">wrote</span> <span m="646650">it</span> <span m="646730">this</span> <span m="646950">way</span> <span m="647330">because</span> <span m="647820">this</span> <span m="649200">might</span> <span m="649470">be</span> <span m="649580">familiar.</span> <span m="651350">Does</span> <span m="651620">everybody</span> <span m="651910">remember</span> <span m="652260">that</span> <span m="652520">from--</span> <span m="652760">I</span> <span m="652830">think</span> <span m="653040">it</span> <span m="653110">was</span> <span m="653350">the</span> <span m="654230">second</span> <span m="654660">recitation?</span> <span m="657950">Anybody</span> <span m="658240">remember</span> <span m="658490">the</span> <span m="658570">formula?</span> <span m="659070">What</span> <span m="659230">this</span> <span m="659420">evaluates</span> <span m="659930">to?</span> <span m="662980">The</span> <span m="663110">sum</span> <span m="663470">of</span> <span m="663600">x</span> <span m="663790">to</span> <span m="663870">the</span> <span m="664020">i,</span> <span m="664520">where</span> <span m="665020">i</span> <span m="665250">goes</span> <span m="665610">from</span> <span m="665770">0</span> <span m="666140">to</span> <span m="666220">n</span> <span m="666390">minus</span> <span m="666710">1?</span> <span m="669380">Remember</span> <span m="669580">that?</span> <span m="671170">One</span> <span m="671580">minus</span> <span m="672020">x</span> <span m="672240">to</span> <span m="672340">the</span> <span m="672510">n.</span> <span m="675730">Remember</span> <span m="676190">1</span> <span m="676670">minus</span> <span m="677050">x.</span> <span m="678820">In</span> <span m="678910">the</span> <span m="678980">second</span> <span m="679500">recitation,</span> <span m="680190">I</span> <span m="680250">think,</span> <span m="680550">we</span> <span m="680660">proved</span> <span m="681980">that</span> <span m="682180">this</span> <span m="682590">equals</span> <span m="682930">that.</span> <span m="683550">What</span> <span m="683740">was</span> <span m="683870">the</span> <span m="683950">proof</span> <span m="684210">technique</span> <span m="684590">we</span> <span m="684710">used?</span> <span m="686370">Induction.</span> <span m="687560">OK?</span> </p>
<p><span m="693490">So,</span> <span m="693660">in</span> <span m="693750">fact,</span> <span m="696320">there's</span> <span m="696530">a</span> <span m="696570">theorem</span> <span m="697040">here.</span> <span m="701060">For</span> <span m="701250">all</span> <span m="701590">n</span> <span m="702420">bigger</span> <span m="702650">and</span> <span m="702720">equal</span> <span m="702910">to</span> <span m="702980">1</span> <span m="703420">and</span> <span m="703660">x</span> <span m="703980">not</span> <span m="704230">equal</span> <span m="704540">to</span> <span m="704640">1,</span> <span m="706730">we</span> <span m="706940">proved</span> <span m="708600">the</span> <span m="708700">sum</span> <span m="709070">from</span> <span m="709230">i</span> <span m="709390">equals</span> <span m="709550">0</span> <span m="709820">to</span> <span m="709900">n</span> <span m="710090">minus</span> <span m="710450">1</span> <span m="711250">x</span> <span m="711570">to</span> <span m="711650">the</span> <span m="712260">i</span> <span m="712360">equals</span> <span m="712810">1</span> <span m="713070">minus</span> <span m="713430">x</span> <span m="713640">to</span> <span m="713730">the</span> <span m="713870">n</span> <span m="714810">over</span> <span m="714960">1</span> <span m="715150">minus</span> <span m="715500">x.</span> <span m="717270">And</span> <span m="717370">so</span> <span m="717450">this</span> <span m="717610">is</span> <span m="717700">a</span> <span m="717760">nice,</span> <span m="718030">closed</span> <span m="718400">form.</span> <span m="718870">No</span> <span m="719060">sum</span> <span m="719330">any</span> <span m="719510">more,</span> <span m="719690">just</span> <span m="720440">that,</span> <span m="721810">which</span> <span m="721970">is</span> <span m="722120">nice.</span> </p>
<p><span m="724040">Now,</span> <span m="724250">induction</span> <span m="724990">proved</span> <span m="725450">it</span> <span m="725540">was</span> <span m="725650">the</span> <span m="725770">right</span> <span m="726010">answer.</span> <span m="727960">Once</span> <span m="729060">you</span> <span m="729890">knew</span> <span m="730170">it--</span> <span m="731320">we</span> <span m="731630">gave</span> <span m="731910">it</span> <span m="731990">to</span> <span m="732110">you--</span> <span m="732740">using</span> <span m="733060">induction</span> <span m="733450">to</span> <span m="733520">prove</span> <span m="733770">that</span> <span m="733930">theorem</span> <span m="734240">wasn't</span> <span m="734520">hard.</span> <span m="736100">What</span> <span m="736250">we're</span> <span m="736370">going</span> <span m="736490">to</span> <span m="736550">look</span> <span m="736730">at</span> <span m="736820">doing</span> <span m="737430">this</span> <span m="737730">week</span> <span m="738030">and</span> <span m="738130">next</span> <span m="738450">week</span> <span m="739310">is</span> <span m="739520">figuring</span> <span m="739880">out</span> <span m="740460">how</span> <span m="740700">to</span> <span m="740810">figure</span> <span m="741230">out</span> <span m="741410">this</span> <span m="741580">was</span> <span m="741730">the</span> <span m="741870">answer</span> <span m="742160">in</span> <span m="742230">the</span> <span m="742300">first</span> <span m="742610">place.</span> <span m="743550">Methods</span> <span m="744320">for</span> <span m="744460">doing</span> <span m="744780">that--</span> <span m="745220">to</span> <span m="745320">evaluate</span> <span m="745810">the</span> <span m="745880">sum--</span> <span m="747760">and</span> <span m="748090">there's</span> <span m="748240">a</span> <span m="748300">lot</span> <span m="748560">of</span> <span m="748640">ways</span> <span m="749090">that</span> <span m="749260">you</span> <span m="749370">can</span> <span m="749570">do</span> <span m="750420">that</span> <span m="750620">particular</span> <span m="751100">sum.</span> </p>
<p><span m="752050">Probably</span> <span m="752410">the</span> <span m="752590">easiest</span> <span m="754790">is</span> <span m="755030">known</span> <span m="755260">as</span> <span m="755380">the</span> <span m="755450">perturbation</span> <span m="756160">method.</span> <span m="760050">This</span> <span m="760310">sometimes</span> <span m="760890">works.</span> <span m="762090">Certainly</span> <span m="762420">with</span> <span m="762520">sums</span> <span m="762770">like</span> <span m="762980">that,</span> <span m="763200">it</span> <span m="763280">often</span> <span m="763620">works.</span> <span m="765200">The</span> <span m="765380">idea</span> <span m="765680">is</span> <span m="765760">as</span> <span m="765900">follows.</span> <span m="766460">We're</span> <span m="766470">trying</span> <span m="766780">to</span> <span m="766850">compute</span> <span m="767860">the</span> <span m="767980">sum</span> <span m="768340">S,</span> <span m="768840">which</span> <span m="769020">is</span> <span m="769120">1</span> <span m="769460">plus</span> <span m="769830">x</span> <span m="770750">plus</span> <span m="771110">x</span> <span m="771310">squared</span> <span m="773730">plus</span> <span m="774070">x</span> <span m="774320">to</span> <span m="774400">the</span> <span m="774530">n</span> <span m="774660">minus</span> <span m="774990">1,</span> <span m="777500">and</span> <span m="777650">what</span> <span m="777740">we're</span> <span m="777830">going</span> <span m="777950">to</span> <span m="778030">do</span> <span m="778280">is</span> <span m="778420">perturb</span> <span m="779040">it</span> <span m="779190">a</span> <span m="779270">little</span> <span m="779500">bit</span> <span m="781230">and</span> <span m="781400">then</span> <span m="781800">subtract</span> <span m="782600">to</span> <span m="782680">get</span> <span m="783080">big</span> <span m="783380">cancellation.</span> </p>
<p><span m="785210">In</span> <span m="785370">this</span> <span m="785520">case,</span> <span m="785760">it's</span> <span m="785880">pretty</span> <span m="786120">simple.</span> <span m="786620">We</span> <span m="787540">multiply</span> <span m="788250">the</span> <span m="788350">sum</span> <span m="788680">by</span> <span m="788820">x</span> <span m="789410">to</span> <span m="789520">get</span> <span m="790600">x</span> <span m="791050">plus</span> <span m="791430">x</span> <span m="791660">squared</span> <span m="792210">plus--</span> <span m="798970">I've</span> <span m="799130">defined</span> <span m="799630">S</span> <span m="799830">to</span> <span m="799920">be</span> <span m="800010">that,</span> <span m="800540">x</span> <span m="800820">times</span> <span m="801220">S--</span> <span m="802090">well,</span> <span m="802240">I</span> <span m="802380">get</span> <span m="802780">x</span> <span m="803150">plus</span> <span m="803580">x</span> <span m="803800">squared</span> <span m="804340">and</span> <span m="805200">so</span> <span m="805450">forth,</span> <span m="805850">up</span> <span m="805970">to</span> <span m="806050">x</span> <span m="806240">to</span> <span m="806320">the</span> <span m="806440">n,</span> <span m="808450">and</span> <span m="808660">now</span> <span m="809020">I</span> <span m="809140">can</span> <span m="809320">subtract</span> <span m="810040">one</span> <span m="810230">from</span> <span m="810390">the</span> <span m="810540">other</span> <span m="811790">and</span> <span m="812020">almost</span> <span m="812420">everything</span> <span m="812740">cancels.</span> <span m="816020">So</span> <span m="816170">I</span> <span m="816260">get</span> <span m="816780">1</span> <span m="817150">minus</span> <span m="817550">x</span> <span m="817880">times</span> <span m="818200">S</span> <span m="819360">equals</span> <span m="821240">1.</span> <span m="822220">These</span> <span m="822480">cancel,</span> <span m="822970">cancel,</span> <span m="823800">cancel.</span> <span m="825450">Minus</span> <span m="825860">x</span> <span m="826070">to</span> <span m="826160">the</span> <span m="826320">n.</span> <span m="828750">And</span> <span m="828930">therefore</span> <span m="830520">S</span> <span m="830880">equals</span> <span m="831320">1</span> <span m="831550">minus</span> <span m="831890">x</span> <span m="832200">to</span> <span m="832270">the</span> <span m="832340">n</span> <span m="832726">over</span> <span m="833112">1</span> <span m="833500">minus</span> <span m="833870">x.</span> </p>
<p><span m="837000">So</span> <span m="837090">that's</span> <span m="839220">a</span> <span m="839260">vague</span> <span m="839940">method.</span> <span m="841820">This</span> <span m="842030">gets</span> <span m="842230">used</span> <span m="842500">all</span> <span m="842740">the</span> <span m="842840">time</span> <span m="843900">in</span> <span m="844050">applied</span> <span m="844400">mathematics,</span> <span m="844930">and</span> <span m="845020">they</span> <span m="845080">call</span> <span m="845280">it</span> <span m="845350">the</span> <span m="845420">perturbation</span> <span m="846020">method.</span> <span m="846930">Take</span> <span m="847180">your</span> <span m="847290">sum,</span> <span m="848880">wiggle</span> <span m="849050">it</span> <span m="849220">around</span> <span m="849530">a</span> <span m="849570">little</span> <span m="849790">bit,</span> <span m="850150">get</span> <span m="850340">something</span> <span m="850460">that</span> <span m="850560">looks</span> <span m="850730">close,</span> <span m="851160">subtract</span> <span m="851800">it,</span> <span m="851990">everything</span> <span m="852370">cancels,</span> <span m="852850">life</span> <span m="853100">is</span> <span m="853220">nice,</span> <span m="854310">and</span> <span m="854450">all</span> <span m="854600">of</span> <span m="854670">a</span> <span m="854710">sudden</span> <span m="855060">you've</span> <span m="855180">figured</span> <span m="855440">out</span> <span m="855730">the</span> <span m="855860">answer.</span> </p>
<p><span m="861530">So</span> <span m="862980">getting</span> <span m="863260">back</span> <span m="863490">to</span> <span m="863620">our</span> <span m="863820">annuity</span> <span m="864150">problem,</span> <span m="866930">we</span> <span m="867070">can</span> <span m="867210">plug</span> <span m="867610">that</span> <span m="867840">formula</span> <span m="869690">back</span> <span m="869990">in</span> <span m="870100">here.</span> <span m="871660">So</span> <span m="871850">the</span> <span m="871920">value</span> <span m="872430">of</span> <span m="872510">the</span> <span m="872610">annuity</span> <span m="875010">is</span> <span m="875320">m</span> <span m="876280">times</span> <span m="877580">1</span> <span m="877810">minus</span> <span m="878270">x</span> <span m="878530">to</span> <span m="878630">the</span> <span m="878790">n</span> <span m="879030">over</span> <span m="879210">1</span> <span m="879420">minus</span> <span m="879770">x.</span> <span m="881080">We'll</span> <span m="881270">plug</span> <span m="881675">in</span> <span m="882080">x</span> <span m="882410">equals</span> <span m="884320">1</span> <span m="884660">over</span> <span m="884880">1</span> <span m="885140">plus</span> <span m="885460">p,</span> <span m="887570">and</span> <span m="887800">we</span> <span m="887930">get</span> <span m="889450">m</span> <span m="890642">1</span> <span m="891050">minus</span> <span m="891590">1</span> <span m="891860">over</span> <span m="892120">1</span> <span m="892440">plus</span> <span m="892760">p</span> <span m="894670">to</span> <span m="894750">the</span> <span m="894960">n</span> <span m="896330">over</span> <span m="896660">1</span> <span m="896950">minus</span> <span m="897330">1</span> <span m="897550">over</span> <span m="897740">1</span> <span m="897970">plus</span> <span m="898300">p,</span> <span m="899870">just</span> <span m="900050">plugging</span> <span m="900440">in.</span> <span m="901830">And</span> <span m="902080">now</span> <span m="902380">to</span> <span m="902490">simplify</span> <span m="903200">this,</span> <span m="903510">I'll</span> <span m="903630">multiply</span> <span m="904160">the</span> <span m="904260">top</span> <span m="904650">and</span> <span m="904780">bottom</span> <span m="905150">by</span> <span m="905290">1</span> <span m="905520">plus</span> <span m="905770">p,</span> <span m="908970">and</span> <span m="909160">I'll</span> <span m="909220">get</span> <span m="909420">1</span> <span m="909660">plus</span> <span m="909870">p</span> <span m="910030">minus</span> <span m="911110">1</span> <span m="911450">on</span> <span m="911510">the</span> <span m="911570">bottom.</span> <span m="911920">Just</span> <span m="912070">gives</span> <span m="912210">me</span> <span m="912310">a</span> <span m="912460">p</span> <span m="912610">on</span> <span m="912720">the</span> <span m="912790">bottom.</span> <span m="914080">I</span> <span m="914140">have</span> <span m="914200">a</span> <span m="914260">1</span> <span m="914460">plus</span> <span m="914710">p</span> <span m="914880">on</span> <span m="914970">the</span> <span m="915040">top</span> <span m="916380">minus</span> <span m="918160">1</span> <span m="918520">over</span> <span m="918790">1</span> <span m="919040">plus</span> <span m="919370">p</span> <span m="921030">to</span> <span m="921150">the</span> <span m="921280">n</span> <span m="921470">minus</span> <span m="921880">1,</span> <span m="922860">all</span> <span m="923200">over</span> <span m="923400">p.</span> </p>
<p><span m="927740">So</span> <span m="927850">now</span> <span m="928160">we</span> <span m="928300">have</span> <span m="928670">a</span> <span m="928770">formula--</span> <span m="929460">closed</span> <span m="929930">form</span> <span m="930230">expression</span> <span m="930850">formula--</span> <span m="931840">for</span> <span m="931980">the</span> <span m="932060">value</span> <span m="932510">of</span> <span m="932580">the</span> <span m="932680">annuity.</span> <span m="933840">All's</span> <span m="934040">we've</span> <span m="934240">got</span> <span m="934390">to</span> <span m="934430">plug</span> <span m="934665">in</span> <span m="934900">is</span> <span m="935500">m,</span> <span m="936090">the</span> <span m="936200">payment</span> <span m="936630">every</span> <span m="936870">year,</span> <span m="937400">n,</span> <span m="937710">the</span> <span m="937770">number</span> <span m="938060">of</span> <span m="938150">years,</span> <span m="939200">and</span> <span m="939350">then</span> <span m="939510">p,</span> <span m="939880">the</span> <span m="940010">interest</span> <span m="940330">rate.</span> <span m="941560">And</span> <span m="941630">so,</span> <span m="941870">for</span> <span m="942180">example,</span> <span m="946080">if</span> <span m="946340">we</span> <span m="946440">made</span> <span m="947370">m</span> <span m="947830">be</span> <span m="947940">$50,000,</span> <span m="951540">as</span> <span m="951770">in</span> <span m="951830">the</span> <span m="951910">lottery.</span> <span m="952390">We</span> <span m="952500">made</span> <span m="952760">n</span> <span m="953130">be</span> <span m="953240">20</span> <span m="953600">years,</span> <span m="955580">and</span> <span m="955880">say</span> <span m="956380">we</span> <span m="956500">took</span> <span m="956730">6%</span> <span m="957360">interest,</span> <span m="958420">which</span> <span m="958640">is</span> <span m="958760">actually</span> <span m="959060">very</span> <span m="959360">good</span> <span m="959560">these</span> <span m="959760">days,</span> <span m="962050">and</span> <span m="962170">I</span> <span m="962230">plug</span> <span m="962590">those</span> <span m="962820">in</span> <span m="962980">here,</span> <span m="963710">the</span> <span m="963830">value</span> <span m="964330">is</span> <span m="964440">going</span> <span m="964570">to</span> <span m="964640">be</span> <span m="965960">$607,906.</span> </p>
<p><span m="972710">All</span> <span m="972920">right.</span> <span m="973210">So</span> <span m="973330">those</span> <span m="973650">of</span> <span m="973750">you</span> <span m="973960">that</span> <span m="974460">preferred</span> <span m="975520">700</span> <span m="976260">grand--</span> <span m="976700">if</span> <span m="976780">you</span> <span m="976880">assume</span> <span m="977160">6%</span> <span m="977680">interest--</span> <span m="978420">you're</span> <span m="978600">right.</span> <span m="979560">Those</span> <span m="979860">of</span> <span m="979885">you</span> <span m="979910">who</span> <span m="980050">preferred</span> <span m="980440">500</span> <span m="980960">grand,</span> <span m="981470">no,</span> <span m="981920">you're</span> <span m="982080">better</span> <span m="982280">off</span> <span m="982450">waiting</span> <span m="983310">and</span> <span m="983450">getting</span> <span m="983650">your</span> <span m="983740">50</span> <span m="984030">grand</span> <span m="984310">a</span> <span m="984370">year.</span> </p>
<p><span m="985360">Now,</span> <span m="985530">of</span> <span m="985600">course,</span> <span m="985900">if</span> <span m="986000">the</span> <span m="986120">interest</span> <span m="986460">rate</span> <span m="987110">is</span> <span m="987310">lower,</span> <span m="988500">well,</span> <span m="988910">that</span> <span m="989250">changes</span> <span m="989700">things.</span> <span m="991020">That</span> <span m="991340">shifts</span> <span m="991810">it</span> <span m="991945">even</span> <span m="992080">more.</span> <span m="992800">The</span> <span m="993230">annuity</span> <span m="993570">is</span> <span m="993670">worth</span> <span m="993860">even</span> <span m="994080">more</span> <span m="994430">if</span> <span m="994520">the</span> <span m="994630">interest</span> <span m="994930">rate</span> <span m="995070">is</span> <span m="995200">lower--</span> <span m="996360">if</span> <span m="996480">p</span> <span m="996650">is</span> <span m="996780">smaller.</span> <span m="998540">In</span> <span m="998690">fact,</span> <span m="998970">say</span> <span m="999250">p</span> <span m="999510">was</span> <span m="999670">0.</span> <span m="1002400">Say</span> <span m="1002670">the</span> <span m="1002740">interest</span> <span m="1003150">rate</span> <span m="1003300">is</span> <span m="1003390">0,</span> <span m="1003870">so</span> <span m="1004040">$1</span> <span m="1004390">today</span> <span m="1004580">equals</span> <span m="1004770">$1</span> <span m="1005050">tomorrow,</span> <span m="1005310">then</span> <span m="1005510">what</span> <span m="1005760">is</span> <span m="1005920">the</span> <span m="1006180">lottery</span> <span m="1006600">worth?</span> <span m="1008350">A</span> <span m="1008460">million</span> <span m="1008800">dollars.</span> <span m="1010130">And</span> <span m="1010320">the</span> <span m="1010390">bigger</span> <span m="1010720">p</span> <span m="1010970">gets,</span> <span m="1011770">the</span> <span m="1011890">less</span> <span m="1012180">your</span> <span m="1012290">payment</span> <span m="1012660">is</span> <span m="1012760">worth.</span> <span m="1017660">Any</span> <span m="1017840">questions</span> <span m="1018250">about</span> <span m="1018480">that?</span> <span m="1021530">OK.</span> </p>
<p><span m="1023320">What</span> <span m="1023670">if</span> <span m="1024900">you</span> <span m="1025109">were</span> <span m="1025250">paid</span> <span m="1025800">$50,000</span> <span m="1027450">a</span> <span m="1027599">year</span> <span m="1028010">forever--</span> <span m="1029780">you</span> <span m="1029950">live</span> <span m="1030170">forever</span> <span m="1031170">or</span> <span m="1031460">it</span> <span m="1031560">goes</span> <span m="1031890">to</span> <span m="1031980">your</span> <span m="1032160">estate</span> <span m="1032619">and</span> <span m="1032740">your</span> <span m="1032880">heirs,</span> <span m="1033950">$50,000</span> <span m="1035079">a</span> <span m="1035150">year</span> <span m="1035349">forever</span> <span m="1036170">or</span> <span m="1037640">a</span> <span m="1037750">million</span> <span m="1038140">dollars</span> <span m="1038520">today.</span> <span m="1041609">How</span> <span m="1041770">many</span> <span m="1041940">people</span> <span m="1042250">want</span> <span m="1042470">the</span> <span m="1042510">million</span> <span m="1042849">dollars</span> <span m="1043180">today?</span> <span m="1045200">How</span> <span m="1045390">many</span> <span m="1045560">want</span> <span m="1045880">50</span> <span m="1046319">grand</span> <span m="1046730">a</span> <span m="1046829">year</span> <span m="1047150">forever?</span> <span m="1049530">Sounds</span> <span m="1050120">good.</span> <span m="1050730">You</span> <span m="1050860">know,</span> <span m="1051480">that's</span> <span m="1051550">an</span> <span m="1051620">infinite</span> <span m="1052020">amount</span> <span m="1052300">of</span> <span m="1052370">money,</span> <span m="1052700">sort</span> <span m="1053060">of.</span> <span m="1054560">It's</span> <span m="1054820">not</span> <span m="1055070">as</span> <span m="1055190">good</span> <span m="1055340">as</span> <span m="1055460">it</span> <span m="1055580">sounds.</span> <span m="1056190">Let's</span> <span m="1056730">see</span> <span m="1056930">why.</span> </p>
<p><span m="1059290">So</span> <span m="1059500">this</span> <span m="1059700">is</span> <span m="1059820">a</span> <span m="1059890">case</span> <span m="1060350">where</span> <span m="1060750">n</span> <span m="1060970">equals</span> <span m="1061220">infinity,</span> <span m="1065150">and</span> <span m="1065590">so</span> <span m="1065720">I'll</span> <span m="1065810">claim</span> <span m="1066320">that</span> <span m="1066460">if</span> <span m="1066880">n</span> <span m="1067240">equals</span> <span m="1067830">infinity,</span> <span m="1070840">then</span> <span m="1071060">the</span> <span m="1071130">value</span> <span m="1071800">of</span> <span m="1071910">this</span> <span m="1072140">annuity</span> <span m="1074280">is</span> <span m="1074460">just</span> <span m="1075040">m</span> <span m="1075280">times</span> <span m="1075840">1</span> <span m="1076100">plus</span> <span m="1076420">p</span> <span m="1076640">over</span> <span m="1076830">p.</span> <span m="1080100">Let's</span> <span m="1080300">see</span> <span m="1080430">why</span> <span m="1080610">that's</span> <span m="1080860">the</span> <span m="1080930">case.</span> <span m="1082880">You</span> <span m="1082990">know,</span> <span m="1083090">it</span> <span m="1083170">sounds</span> <span m="1083510">hard</span> <span m="1083730">to</span> <span m="1083840">evaluate,</span> <span m="1084330">because</span> <span m="1084510">it's</span> <span m="1084580">an</span> <span m="1084650">infinite</span> <span m="1085010">number</span> <span m="1085210">of</span> <span m="1085290">payments,</span> <span m="1087930">but</span> <span m="1088140">what</span> <span m="1088340">happens</span> <span m="1088870">here</span> <span m="1089200">when</span> <span m="1089390">n</span> <span m="1089610">goes</span> <span m="1089850">to</span> <span m="1089950">infinity?</span> <span m="1092250">What</span> <span m="1092340">happens</span> <span m="1092690">to</span> <span m="1092780">this</span> <span m="1092970">thing?</span> <span m="1095300">That</span> <span m="1095490">goes</span> <span m="1095740">to</span> <span m="1095830">0</span> <span m="1097150">as</span> <span m="1097450">n</span> <span m="1097790">goes</span> <span m="1098040">to</span> <span m="1098160">infinity,</span> <span m="1098530">as</span> <span m="1098620">long</span> <span m="1098780">as</span> <span m="1098890">p</span> <span m="1099090">is</span> <span m="1099230">bigger</span> <span m="1099450">than</span> <span m="1099590">0.</span> </p>
<p><span m="1100000">So</span> <span m="1100180">we're</span> <span m="1100290">going</span> <span m="1100410">to</span> <span m="1100470">assume</span> <span m="1100880">6%</span> <span m="1101490">interest.</span> <span m="1102800">So</span> <span m="1102990">that</span> <span m="1103200">goes</span> <span m="1103420">away,</span> <span m="1103940">so</span> <span m="1103990">the</span> <span m="1104100">annuity</span> <span m="1104460">is</span> <span m="1104590">worth</span> <span m="1105050">just</span> <span m="1105340">that,</span> <span m="1105650">m</span> <span m="1105840">times</span> <span m="1106140">1</span> <span m="1106440">plus</span> <span m="1106720">p</span> <span m="1106930">over</span> <span m="1107120">p,</span> <span m="1109200">because</span> <span m="1109640">the</span> <span m="1110340">limit</span> <span m="1110690">as</span> <span m="1110840">n</span> <span m="1111000">goes</span> <span m="1111240">to</span> <span m="1111350">infinity</span> <span m="1112810">of</span> <span m="1113250">1</span> <span m="1113580">over</span> <span m="1113770">1</span> <span m="1113960">plus</span> <span m="1114270">p</span> <span m="1114840">to</span> <span m="1114930">the</span> <span m="1115050">n</span> <span m="1115180">minus</span> <span m="1115530">1,</span> <span m="1116550">that's</span> <span m="1116860">going</span> <span m="1117120">to</span> <span m="1117200">0.</span> <span m="1119890">So</span> <span m="1120070">the</span> <span m="1120150">value</span> <span m="1122840">for</span> <span m="1123250">m</span> <span m="1123580">is</span> <span m="1123760">$50,000,</span> <span m="1128140">and</span> <span m="1128450">at</span> <span m="1128560">6%</span> <span m="1132860">V</span> <span m="1133190">is</span> <span m="1133420">only</span> <span m="1133800">$883,000.</span> <span m="1142390">So</span> <span m="1142590">you're</span> <span m="1142710">better</span> <span m="1143110">off</span> <span m="1143370">taking</span> <span m="1143660">a</span> <span m="1143710">million</span> <span m="1144060">dollars</span> <span m="1144450">today</span> <span m="1145700">than</span> <span m="1145870">$50,000</span> <span m="1146870">a</span> <span m="1146950">year</span> <span m="1147180">forever.</span> </p>
<p><span m="1148780">Now,</span> <span m="1149080">if</span> <span m="1149240">you</span> <span m="1149330">think</span> <span m="1149620">about</span> <span m="1149940">it,</span> <span m="1150060">and</span> <span m="1150190">think</span> <span m="1150370">about</span> <span m="1150570">it</span> <span m="1150620">as</span> <span m="1150670">an</span> <span m="1150770">interest</span> <span m="1151150">rate,</span> <span m="1151790">why</span> <span m="1152090">should</span> <span m="1152270">it</span> <span m="1152350">be</span> <span m="1152660">obvious</span> <span m="1153370">that</span> <span m="1153470">you're</span> <span m="1153590">better</span> <span m="1153930">off</span> <span m="1154160">with</span> <span m="1154270">a</span> <span m="1154310">million</span> <span m="1154620">dollars</span> <span m="1154950">today</span> <span m="1156220">than</span> <span m="1156390">50</span> <span m="1156700">grand</span> <span m="1157000">a</span> <span m="1157080">year</span> <span m="1157250">forever?</span> <span m="1159180">Think</span> <span m="1159400">about</span> <span m="1159680">what</span> <span m="1159860">you</span> <span m="1159960">could</span> <span m="1160130">do</span> <span m="1160340">with</span> <span m="1160460">that</span> <span m="1160640">million</span> <span m="1160930">dollars</span> <span m="1161470">if</span> <span m="1161570">you</span> <span m="1161690">had</span> <span m="1161870">it</span> <span m="1161950">today</span> <span m="1163990">at</span> <span m="1164170">6%</span> <span m="1164910">interest.</span> <span m="1167790">What</span> <span m="1168020">would</span> <span m="1168130">you</span> <span m="1168210">do</span> <span m="1168420">with</span> <span m="1168570">it</span> <span m="1168680">to</span> <span m="1169070">make</span> <span m="1169320">more</span> <span m="1169560">money</span> <span m="1169880">than</span> <span m="1171130">50</span> <span m="1171420">grand</span> <span m="1171710">a</span> <span m="1171790">year</span> <span m="1171950">forever?</span> <span m="1172360">Yeah.</span> </p>
<p><span m="1173752">AUDIENCE: You</span> <span m="1174216">could</span> <span m="1174293">have</span> <span m="1174370">like--</span> <span m="1174448">you</span> <span m="1174525">could</span> <span m="1174602">make</span> <span m="1174680">$50,000</span> <span m="1175610">per</span> <span m="1175856">year</span> <span m="1176102">just off of</span> <span m="1176594">[INAUDIBLE].</span> </p>
<p><span m="1178070">PROFESSOR: Yeah.</span> <span m="1178420">In</span> <span m="1178540">this</span> <span m="1178740">model,</span> <span m="1179480">if</span> <span m="1179720">the</span> <span m="1179820">interest</span> <span m="1180030">rate</span> <span m="1180150">is</span> <span m="1180230">6%,</span> <span m="1180850">you</span> <span m="1180930">can</span> <span m="1181030">put</span> <span m="1181210">in</span> <span m="1181270">the</span> <span m="1181340">bank.</span> <span m="1181820">It</span> <span m="1182160">makes</span> <span m="1182440">6%</span> <span m="1182930">every</span> <span m="1183100">year,</span> <span m="1183220">that's</span> <span m="1183370">60</span> <span m="1183740">grand</span> <span m="1184060">a</span> <span m="1184140">year</span> <span m="1184370">forever.</span> <span m="1185300">Better</span> <span m="1185570">than</span> <span m="1185710">50</span> <span m="1186020">grand</span> <span m="1186280">a</span> <span m="1186340">year</span> <span m="1186750">forever.</span> <span m="1188150">So</span> <span m="1188290">maybe</span> <span m="1188510">it's--</span> <span m="1189456">even</span> <span m="1189648">without</span> <span m="1189840">doing</span> <span m="1190000">the</span> <span m="1190070">math,</span> <span m="1190340">you</span> <span m="1190440">can</span> <span m="1190550">tell</span> <span m="1190720">which</span> <span m="1190890">way</span> <span m="1191000">it's</span> <span m="1191090">going</span> <span m="1191210">to</span> <span m="1191270">go,</span> <span m="1191960">but</span> <span m="1192090">this</span> <span m="1192280">tells</span> <span m="1192610">you</span> <span m="1193070">exactly</span> <span m="1193710">what</span> <span m="1193850">it's</span> <span m="1193990">worth.</span> <span m="1198460">Any</span> <span m="1198640">questions?</span> <span m="1202340">OK.</span> <span m="1204380">Yeah.</span> </p>
<p><span m="1204862">AUDIENCE:</span> <span m="1205344">[INAUDIBLE]</span> <span m="1205826">current</span> <span m="1206308">value--</span> <span m="1206428">that's</span> <span m="1206549">how</span> <span m="1206669">much</span> <span m="1206790">it's</span> <span m="1207272">worth</span> <span m="1207754">to</span> <span m="1207874">you</span> <span m="1207995">right</span> <span m="1208115">now.</span> </p>
<p><span m="1208720">PROFESSOR: Yeah.</span> </p>
<p><span m="1209220">AUDIENCE:  So</span> <span m="1209720">and,</span> <span m="1211220">like,</span> <span m="1212220">if you</span> <span m="1212720">have</span> <span m="1212970">$1</span> <span m="1213220">in</span> <span m="1213386">the</span> <span m="1213553">future--</span> </p>
<p><span m="1214010">PROFESSOR: Yeah.</span> </p>
<p><span m="1215086">AUDIENCE: --where</span> <span m="1215534">S</span> <span m="1215758">right</span> <span m="1215982">now</span> <span m="1216430">is</span> <span m="1216878">m</span> <span m="1216967">over</span> <span m="1217057">1</span> <span m="1217146">plus</span> <span m="1217236">p--</span> </p>
<p><span m="1218230">PROFESSOR:  Yeah.</span> </p>
<p><span m="1218535">AUDIENCE:</span> <span m="1218611">--is</span> <span m="1218687">it</span> <span m="1218763">worth</span> <span m="1218840">less</span> <span m="1218994">to</span> <span m="1219148">you</span> <span m="1219302">[INAUDIBLE]</span> <span m="1219764">then</span> <span m="1220226">later</span> <span m="1220688">on,</span> <span m="1220919">it's</span> <span m="1221150">going</span> <span m="1221304">to</span> <span m="1221458">be</span> <span m="1221612">worth</span> <span m="1221727">more</span> <span m="1221843">to</span> <span m="1221958">you.</span> </p>
<p><span m="1223000">PROFESSOR: Ah,</span> <span m="1223150">so</span> <span m="1223740">if</span> <span m="1223890">you</span> <span m="1224020">move</span> <span m="1224370">yourself</span> <span m="1224930">forward</span> <span m="1225220">in</span> <span m="1225330">time,</span> <span m="1225870">$1</span> <span m="1226440">a</span> <span m="1226560">year,</span> <span m="1226980">in</span> <span m="1227230">a</span> <span m="1227480">year</span> <span m="1227535">it'll</span> <span m="1227590">worth</span> <span m="1227810">$1</span> <span m="1228197">to</span> <span m="1228390">you--</span> </p>
<p><span m="1228584">AUDIENCE: Yeah.</span> </p>
<p><span m="1229360">PROFESSOR: --but</span> <span m="1229550">today</span> <span m="1231030">it's</span> <span m="1231240">worth</span> <span m="1231540">less</span> <span m="1231810">than</span> <span m="1231910">$1</span> <span m="1232290">to</span> <span m="1232420">you,</span> <span m="1232490">because</span> <span m="1232640">you</span> <span m="1232750">could</span> <span m="1232900">take</span> <span m="1234020">the</span> <span m="1234140">dollar</span> <span m="1234460">today</span> <span m="1234690">and</span> <span m="1234780">invest</span> <span m="1235070">in</span> <span m="1235150">the</span> <span m="1235230">bank,</span> <span m="1235520">and</span> <span m="1235620">it's</span> <span m="1235910">worth</span> <span m="1236190">more</span> <span m="1236500">in</span> <span m="1236570">a</span> <span m="1236630">year,</span> <span m="1237680">because</span> <span m="1237850">the</span> <span m="1237910">money</span> <span m="1238180">grows</span> <span m="1238570">in</span> <span m="1238680">value,</span> <span m="1239950">as</span> <span m="1240060">a</span> <span m="1240090">way</span> <span m="1240200">to</span> <span m="1240270">think</span> <span m="1240490">of</span> <span m="1240580">it,</span> <span m="1241700">because</span> <span m="1241890">you</span> <span m="1241980">can</span> <span m="1242190">earn</span> <span m="1242360">interest</span> <span m="1242680">on</span> <span m="1242820">it.</span> <span m="1243510">What's</span> <span m="1243690">that?</span> </p>
<p><span m="1244104">AUDIENCE:</span> <span m="1244518">You</span> <span m="1244725">could</span> <span m="1244932">spend</span> <span m="1245139">it.</span> </p>
<p><span m="1245760">PROFESSOR: Yeah.</span> <span m="1246480">Yeah,</span> <span m="1246850">if</span> <span m="1246910">you</span> <span m="1246970">just</span> <span m="1247030">spend</span> <span m="1247380">it,</span> <span m="1247680">well,</span> <span m="1248030">then</span> <span m="1248170">at</span> <span m="1248270">least</span> <span m="1248470">you</span> <span m="1248550">had</span> <span m="1248690">the</span> <span m="1248790">use</span> <span m="1249110">of</span> <span m="1249210">what</span> <span m="1249360">you</span> <span m="1249440">spent</span> <span m="1249690">it</span> <span m="1249770">on</span> <span m="1249910">for</span> <span m="1250030">the</span> <span m="1250130">year.</span> <span m="1251100">So</span> <span m="1251150">there's</span> <span m="1251300">some</span> <span m="1251470">other</span> <span m="1251670">kind</span> <span m="1251910">of</span> <span m="1251980">value,</span> <span m="1252500">right?</span> <span m="1252800">Maybe</span> <span m="1252940">you</span> <span m="1253400">bought</span> <span m="1253640">a</span> <span m="1253700">house</span> <span m="1254210">or</span> <span m="1254950">something</span> <span m="1255660">that--</span> <span m="1256310">maybe</span> <span m="1256540">something</span> <span m="1256750">that</span> <span m="1256840">even</span> <span m="1257020">appreciated</span> <span m="1257560">in</span> <span m="1257640">value.</span> <span m="1259410">OK.</span> </p>
<p><span m="1260610">But</span> <span m="1261000">these</span> <span m="1261210">things</span> <span m="1261480">get</span> <span m="1262750">sort</span> <span m="1263370">of</span> <span m="1263440">squishy,</span> <span m="1264650">and</span> <span m="1264800">that</span> <span m="1264940">is</span> <span m="1265170">where</span> <span m="1265360">a</span> <span m="1265420">lot</span> <span m="1265530">of</span> <span m="1265590">people</span> <span m="1265770">make</span> <span m="1266290">money</span> <span m="1266520">on</span> <span m="1266610">Wall</span> <span m="1266780">Street,</span> <span m="1267510">is</span> <span m="1267700">because</span> <span m="1268170">different</span> <span m="1268460">companies</span> <span m="1268890">have</span> <span m="1269030">different</span> <span m="1269330">needs</span> <span m="1269940">for</span> <span m="1270060">money.</span> <span m="1270330">They</span> <span m="1270410">have</span> <span m="1270500">different</span> <span m="1270780">views</span> <span m="1271040">of</span> <span m="1271100">what</span> <span m="1271220">the</span> <span m="1271310">interest</span> <span m="1271630">rates</span> <span m="1271820">are</span> <span m="1271860">going</span> <span m="1271980">to</span> <span m="1272040">be,</span> <span m="1272190">and</span> <span m="1272290">you</span> <span m="1272360">can</span> <span m="1272470">play</span> <span m="1272700">in</span> <span m="1272770">the</span> <span m="1272830">middle</span> <span m="1274180">and</span> <span m="1274820">make</span> <span m="1275040">a</span> <span m="1275090">lot</span> <span m="1275250">of</span> <span m="1275310">money</span> <span m="1275500">that</span> <span m="1275720">way.</span> </p>
<p><span m="1295220">So</span> <span m="1295850">more</span> <span m="1296030">generally,</span> <span m="1296840">there's</span> <span m="1297030">a</span> <span m="1297070">corollary</span> <span m="1297630">to</span> <span m="1297720">the</span> <span m="1297800">theorem,</span> <span m="1299620">and</span> <span m="1299710">that</span> <span m="1299910">is</span> <span m="1300290">that</span> <span m="1300810">if</span> <span m="1300960">the</span> <span m="1301080">absolute</span> <span m="1301520">value</span> <span m="1301790">of</span> <span m="1301880">x</span> <span m="1302150">is</span> <span m="1302290">less</span> <span m="1302600">than</span> <span m="1302730">1,</span> <span m="1304030">then</span> <span m="1304210">the</span> <span m="1304280">sum</span> <span m="1305960">i</span> <span m="1306350">equals</span> <span m="1306500">0</span> <span m="1308100">to</span> <span m="1308250">infinity</span> <span m="1309820">x</span> <span m="1310080">to</span> <span m="1310180">the</span> <span m="1310330">i</span> <span m="1311650">is</span> <span m="1311870">just</span> <span m="1312210">1</span> <span m="1312440">over</span> <span m="1312650">1</span> <span m="1312870">minus</span> <span m="1313220">x.</span> <span m="1316080">We</span> <span m="1316180">didn't</span> <span m="1316470">prove</span> <span m="1316800">this</span> <span m="1317920">back</span> <span m="1318210">in</span> <span m="1318290">the</span> <span m="1318360">second</span> <span m="1318700">recitation,</span> <span m="1319350">because</span> <span m="1320030">there's</span> <span m="1320220">no</span> <span m="1320420">n</span> <span m="1320710">to</span> <span m="1320830">induct</span> <span m="1321240">on</span> <span m="1321390">here,</span> <span m="1322130">but</span> <span m="1322300">the</span> <span m="1322390">proof</span> <span m="1322680">is</span> <span m="1322790">simple</span> <span m="1323320">from</span> <span m="1323510">the</span> <span m="1323570">theorem.</span> <span m="1325220">And</span> <span m="1325400">it's</span> <span m="1325500">simply</span> <span m="1325770">because</span> <span m="1326900">if</span> <span m="1327070">x</span> <span m="1327470">is</span> <span m="1327600">less</span> <span m="1327820">than</span> <span m="1327920">1,</span> <span m="1328060">an</span> <span m="1328150">absolute</span> <span m="1328550">value,</span> <span m="1329340">as</span> <span m="1329550">n</span> <span m="1329740">goes</span> <span m="1329970">to</span> <span m="1330070">infinity,</span> <span m="1331510">that</span> <span m="1331750">goes</span> <span m="1332010">to</span> <span m="1332100">0,</span> <span m="1333330">and</span> <span m="1333490">so</span> <span m="1333560">you're</span> <span m="1333620">just</span> <span m="1333890">left</span> <span m="1334100">with</span> <span m="1334210">1</span> <span m="1334410">over</span> <span m="1334550">1</span> <span m="1334720">minus</span> <span m="1335040">x.</span> </p>
<p><span m="1346650">So,</span> <span m="1346990">for</span> <span m="1347130">example,</span> <span m="1350770">what's</span> <span m="1351210">this</span> <span m="1351500">sum?</span> <span m="1351680">This</span> <span m="1351830">one</span> <span m="1351990">you</span> <span m="1352090">all</span> <span m="1352280">know,</span> <span m="1352500">I'm</span> <span m="1352620">sure.</span> <span m="1357010">Out</span> <span m="1357220">to</span> <span m="1357330">infinity.</span> <span m="1357880">What's</span> <span m="1358000">that</span> <span m="1358190">sum</span> <span m="1358440">to?</span> <span m="1360970">To</span> <span m="1361160">2.</span> <span m="1361600">Yeah.</span> <span m="1362250">It's</span> <span m="1362860">1</span> <span m="1363180">over</span> <span m="1363450">1</span> <span m="1363720">minus</span> <span m="1364110">1/2,</span> <span m="1365480">which</span> <span m="1365700">is</span> <span m="1365820">2.</span> <span m="1367710">What</span> <span m="1367940">about</span> <span m="1368380">this</span> <span m="1368849">sum</span> <span m="1374740">out</span> <span m="1374940">to</span> <span m="1375030">infinity?</span> <span m="1375370">What</span> <span m="1375520">does</span> <span m="1375600">that</span> <span m="1375780">sum</span> <span m="1376040">to?</span> </p>
<p><span m="1379263">AUDIENCE:</span> <span m="1379696">[INAUDIBLE]</span> </p>
<p><span m="1381430">PROFESSOR: Yeah,</span> <span m="1382090">3/2,</span> <span m="1383400">1 over</span> <span m="1383650">1</span> <span m="1383890">minus</span> <span m="1384240">1/3</span> <span m="1384738">is</span> <span m="1385236">3/2.</span> <span m="1387700">So,</span> <span m="1387820">easy</span> <span m="1388150">corollaries.</span> <span m="1389920">These</span> <span m="1390150">are</span> <span m="1390210">all</span> <span m="1390410">examples</span> <span m="1390910">of</span> <span m="1390990">geometric</span> <span m="1391660">series.</span> <span m="1392460">That's</span> <span m="1392680">what</span> <span m="1392780">a</span> <span m="1392820">definition</span> <span m="1393330">of</span> <span m="1393390">a</span> <span m="1393430">geometric</span> <span m="1393950">series</span> <span m="1394400">is.</span> <span m="1394600">Something</span> <span m="1394880">that's</span> <span m="1395000">going</span> <span m="1395260">down</span> <span m="1395480">by</span> <span m="1395620">a</span> <span m="1395670">fixed--</span> <span m="1396410">each</span> <span m="1396620">term</span> <span m="1396810">goes</span> <span m="1396970">down</span> <span m="1397130">by</span> <span m="1397420">the</span> <span m="1397540">same</span> <span m="1397820">fixed</span> <span m="1398120">amount</span> <span m="1398360">every</span> <span m="1398520">time.</span> <span m="1399490">And</span> <span m="1399700">geometric</span> <span m="1400250">series,</span> <span m="1401310">generally</span> <span m="1401850">sum</span> <span m="1402580">to</span> <span m="1403440">something</span> <span m="1403830">that</span> <span m="1404190">is</span> <span m="1404350">very</span> <span m="1404550">close</span> <span m="1404910">to</span> <span m="1404990">the</span> <span m="1405070">largest</span> <span m="1405550">term.</span> <span m="1406140">In</span> <span m="1406280">this</span> <span m="1406430">case,</span> <span m="1406730">it's</span> <span m="1407030">1.</span> <span m="1408590">Very</span> <span m="1408870">common,</span> <span m="1410410">because</span> <span m="1410730">of</span> <span m="1410790">that</span> <span m="1410940">formula.</span> <span m="1411380">It's</span> <span m="1411495">1</span> <span m="1411610">over</span> <span m="1411740">1</span> <span m="1411950">minus</span> <span m="1412280">x.</span> <span m="1420270">Any</span> <span m="1420450">questions</span> <span m="1421280">about</span> <span m="1422400">this</span> <span m="1422710">or</span> <span m="1422770">geometric</span> <span m="1423220">series?</span> <span m="1426760">All</span> <span m="1426820">right.</span> </p>
<p><span m="1428280">Well,</span> <span m="1429220">those</span> <span m="1429550">are</span> <span m="1429640">straight</span> <span m="1430100">geometric</span> <span m="1430680">sums.</span> <span m="1431140">Sometimes</span> <span m="1431810">you</span> <span m="1431930">run</span> <span m="1432130">into</span> <span m="1432280">things</span> <span m="1432650">that</span> <span m="1432750">are</span> <span m="1432790">a</span> <span m="1432840">little</span> <span m="1433140">bit</span> <span m="1433300">more</span> <span m="1433500">complicated.</span> <span m="1434930">For</span> <span m="1435050">example,</span> <span m="1436900">say</span> <span m="1437170">I</span> <span m="1437250">have</span> <span m="1437650">this</span> <span m="1437830">kind</span> <span m="1438010">of</span> <span m="1438070">a</span> <span m="1438110">sum,</span> <span m="1438670">i</span> <span m="1438980">equals</span> <span m="1439400">1</span> <span m="1439700">to</span> <span m="1439850">n,</span> <span m="1441570">i</span> <span m="1441930">times</span> <span m="1442650">x</span> <span m="1442920">to</span> <span m="1443030">the</span> <span m="1443130">i.</span> <span m="1445030">Now,</span> <span m="1445145">those</span> <span m="1445260">are</span> <span m="1445360">adding</span> <span m="1445740">up</span> <span m="1445990">x</span> <span m="1446470">plus</span> <span m="1446790">2x</span> <span m="1447350">squared</span> <span m="1448430">plus</span> <span m="1448710">3x</span> <span m="1449340">cubed,</span> <span m="1451100">and</span> <span m="1451210">so</span> <span m="1451450">forth,</span> <span m="1452420">up</span> <span m="1452640">to</span> <span m="1453000">n</span> <span m="1453560">x</span> <span m="1453850">to</span> <span m="1453960">the</span> <span m="1454110">n.</span> <span m="1456780">You</span> <span m="1456870">know,</span> <span m="1456950">that's</span> <span m="1457150">a</span> <span m="1457200">little</span> <span m="1457410">more</span> <span m="1457550">complicated.</span> <span m="1458160">The</span> <span m="1458220">terms</span> <span m="1458680">are</span> <span m="1458730">getting--</span> <span m="1460610">decreasing</span> <span m="1461480">by</span> <span m="1461630">a</span> <span m="1461690">factor</span> <span m="1462010">of</span> <span m="1462100">x,</span> <span m="1462520">increasing</span> <span m="1463090">by</span> <span m="1463230">1</span> <span m="1463620">in</span> <span m="1463700">terms</span> <span m="1463940">of</span> <span m="1464000">the</span> <span m="1464060">coefficient</span> <span m="1464680">every</span> <span m="1464880">time.</span> <span m="1465420">A</span> <span m="1465450">little</span> <span m="1465630">trickier.</span> </p>
<p><span m="1467200">So</span> <span m="1467240">say</span> <span m="1467510">we</span> <span m="1467630">wanted</span> <span m="1467910">to</span> <span m="1468030">get</span> <span m="1468220">a</span> <span m="1468280">closed</span> <span m="1468910">form</span> <span m="1469210">expression</span> <span m="1469760">for</span> <span m="1469890">that?</span> <span m="1471280">There</span> <span m="1471390">are</span> <span m="1471430">several</span> <span m="1471760">ways</span> <span m="1472140">we</span> <span m="1472240">can</span> <span m="1472410">do</span> <span m="1472620">it.</span> <span m="1473130">The</span> <span m="1473270">first</span> <span m="1473680">would</span> <span m="1473790">be</span> <span m="1473890">to</span> <span m="1473990">try</span> <span m="1474220">to</span> <span m="1474360">use</span> <span m="1475370">perturbation--</span> <span m="1476450">the</span> <span m="1476590">perturbation</span> <span m="1477190">method.</span> <span m="1477900">Let's</span> <span m="1478560">try</span> <span m="1478860">that.</span> <span m="1480730">So</span> <span m="1480810">we</span> <span m="1480890">write</span> <span m="1481220">S</span> <span m="1481600">equals</span> <span m="1482090">x</span> <span m="1482490">plus</span> <span m="1482770">2x</span> <span m="1483260">squared</span> <span m="1483880">plus</span> <span m="1484150">3x</span> <span m="1484930">cubed</span> <span m="1486990">plus</span> <span m="1487460">nx</span> <span m="1488370">to</span> <span m="1488460">the</span> <span m="1488630">n,</span> <span m="1490220">and</span> <span m="1490400">let's</span> <span m="1490550">try</span> <span m="1490780">the</span> <span m="1490870">same</span> <span m="1491170">perturbation.</span> <span m="1491920">Multiply</span> <span m="1492420">by</span> <span m="1492610">x,</span> <span m="1494960">I</span> <span m="1495150">get</span> <span m="1495550">that</span> <span m="1496190">x</span> <span m="1496470">squared</span> <span m="1497610">plus</span> <span m="1497910">2x</span> <span m="1498377">cubed</span> <span m="1501180">plus</span> <span m="1501710">n</span> <span m="1501980">minus</span> <span m="1502380">1x</span> <span m="1503190">to</span> <span m="1503270">the</span> <span m="1503440">n,</span> <span m="1504280">plus</span> <span m="1505050">nx</span> <span m="1506240">to</span> <span m="1506300">the</span> <span m="1506640">n</span> <span m="1506770">plus</span> <span m="1506900">1.</span> <span m="1509420">And</span> <span m="1509730">then</span> <span m="1509970">I</span> <span m="1510300">subtract</span> <span m="1511190">to</span> <span m="1511340">try</span> <span m="1511510">to</span> <span m="1511590">get</span> <span m="1511720">all</span> <span m="1511830">the</span> <span m="1511900">cancellation.</span> </p>
<p><span m="1515220">So</span> <span m="1515460">then</span> <span m="1515580">I</span> <span m="1516970">do</span> <span m="1517160">that.</span> <span m="1517410">I</span> <span m="1517440">get</span> <span m="1517600">1</span> <span m="1517790">minus</span> <span m="1518160">x</span> <span m="1518450">times</span> <span m="1518830">S.</span> <span m="1521230">Well,</span> <span m="1521400">I</span> <span m="1521450">didn't</span> <span m="1521660">quite</span> <span m="1522320">cancel</span> <span m="1523100">everything.</span> <span m="1524540">X</span> <span m="1525020">plus</span> <span m="1525920">x</span> <span m="1526290">squared</span> <span m="1527840">plus</span> <span m="1528420">x</span> <span m="1528520">cubed</span> <span m="1531240">plus</span> <span m="1531670">x</span> <span m="1531960">to</span> <span m="1532050">the</span> <span m="1532220">n</span> <span m="1533120">plus</span> <span m="1533640">n--</span> <span m="1534610">or</span> <span m="1534740">minus</span> <span m="1535750">nx</span> <span m="1536340">to</span> <span m="1536430">the</span> <span m="1536750">n</span> <span m="1536880">plus</span> <span m="1537010">1.</span> <span m="1537960">Ah,</span> <span m="1538310">it</span> <span m="1538425">didn't</span> <span m="1538540">quite</span> <span m="1538800">work.</span> <span m="1541010">Anybody</span> <span m="1541380">see</span> <span m="1541590">a</span> <span m="1541660">way</span> <span m="1541880">that</span> <span m="1542010">I</span> <span m="1542120">can</span> <span m="1542660">fix</span> <span m="1542980">this</span> <span m="1543230">up?</span> <span m="1549480">What</span> <span m="1549620">about</span> <span m="1549860">this</span> <span m="1550050">piece?</span> <span m="1551610">That's</span> <span m="1551830">still</span> <span m="1552170">a</span> <span m="1552230">mess</span> <span m="1552560">here.</span> <span m="1553020">Can</span> <span m="1553170">I</span> <span m="1553250">simplify</span> <span m="1553770">that?</span> <span m="1556240">Yeah?</span> </p>
<p><span m="1557522">AUDIENCE:</span> <span m="1557998">[INAUDIBLE]</span> </p>
<p><span m="1565150">PROFESSOR: Yeah.</span> <span m="1567090">There's</span> </p>
<p><span m="1567260">a</span> <span m="1567320">simpler</span> <span m="1567910">way.</span> <span m="1568990">Yeah.</span> </p>
<p><span m="1569882">AUDIENCE:</span> <span m="1570030">That's</span> <span m="1570179">a</span> <span m="1570328">geometric</span> <span m="1570551">series.</span> </p>
<p><span m="1571220">PROFESSOR: That's</span> <span m="1571430">a</span> <span m="1571470">geometric</span> <span m="1571950">series.</span> <span m="1573160">We</span> <span m="1573240">just</span> <span m="1573530">got</span> <span m="1573710">the</span> <span m="1573780">formula</span> <span m="1574280">for</span> <span m="1574590">it,</span> <span m="1574900">so</span> <span m="1575000">that's</span> <span m="1575280">easy.</span> <span m="1576360">This</span> <span m="1577490">equals</span> <span m="1579790">1</span> <span m="1580130">minus</span> <span m="1580560">x</span> <span m="1580820">to</span> <span m="1580910">the</span> <span m="1581060">n</span> <span m="1581400">over</span> <span m="1581710">1</span> <span m="1582010">minus</span> <span m="1582410">x</span> <span m="1583640">minus</span> <span m="1584130">the</span> <span m="1584620">1,</span> <span m="1585140">because</span> <span m="1585300">I'm</span> <span m="1585410">missing</span> <span m="1585750">the</span> <span m="1585830">1</span> <span m="1586090">here.</span> <span m="1586910">So</span> <span m="1587080">we</span> <span m="1587180">can</span> <span m="1587340">rewrite</span> <span m="1587730">this</span> <span m="1587900">whole</span> <span m="1588130">thing</span> <span m="1588400">over</span> <span m="1588570">here.</span> <span m="1599720">Oops.</span> <span m="1600050">Yikes.</span> <span m="1600540">Got</span> <span m="1601010">attacked.</span> <span m="1609880">What's</span> <span m="1610080">that?</span> </p>
<p><span m="1610560">AUDIENCE:</span> <span m="1611040">Would it be</span> <span m="1611520">1 minus x</span> <span m="1612000">should be</span> <span m="1612480">n plus</span> <span m="1612973">1?</span> </p>
<p><span m="1613960">PROFESSOR: Yes</span> <span m="1615790">it</span> <span m="1615940">would.</span> <span m="1616710">That's</span> <span m="1616980">right.</span> <span m="1617360">I've</span> <span m="1617420">got</span> <span m="1617540">to</span> <span m="1617600">add</span> <span m="1617810">1</span> <span m="1618010">to</span> <span m="1618120">there.</span> <span m="1619630">OK.</span> <span m="1620200">So</span> <span m="1621100">that's</span> <span m="1621340">good.</span> <span m="1622630">So</span> <span m="1622820">that</span> <span m="1623030">says</span> <span m="1623500">that</span> <span m="1624790">1</span> <span m="1625180">minus</span> <span m="1625690">x</span> <span m="1627210">times</span> <span m="1627530">S</span> <span m="1628040">equals</span> <span m="1628930">1</span> <span m="1629230">minus</span> <span m="1629630">x</span> <span m="1629890">to</span> <span m="1629980">the</span> <span m="1630110">n</span> <span m="1630270">plus</span> <span m="1630610">1</span> <span m="1631520">over</span> <span m="1631720">1</span> <span m="1631930">minus</span> <span m="1632440">x</span> <span m="1633360">minus</span> <span m="1633870">1,</span> <span m="1635320">and</span> <span m="1635460">then</span> <span m="1635570">I've</span> <span m="1635620">got</span> <span m="1635770">to</span> <span m="1635830">remember</span> <span m="1636100">to</span> <span m="1636390">subtract</span> <span m="1636850">that</span> <span m="1637050">term</span> <span m="1637250">too.</span> <span m="1637650">Minus</span> <span m="1638020">nx</span> <span m="1639020">to</span> <span m="1639150">the</span> <span m="1639280">n</span> <span m="1639410">plus</span> <span m="1639700">1.</span> <span m="1641630">That</span> <span m="1641920">means</span> <span m="1642310">now</span> <span m="1642490">I</span> <span m="1642570">just</span> <span m="1642750">divide</span> <span m="1643140">through,</span> <span m="1644700">and</span> <span m="1644910">I</span> <span m="1645010">simplify--</span> <span m="1645770">divide</span> <span m="1646120">through</span> <span m="1646230">by</span> <span m="1646360">1</span> <span m="1646540">minus</span> <span m="1646840">x--</span> <span m="1647050">and</span> <span m="1647160">simplify,</span> <span m="1647990">and</span> <span m="1648110">I</span> <span m="1648170">get</span> <span m="1648340">the</span> <span m="1648410">following</span> <span m="1648810">formula.</span> <span m="1650070">I</span> <span m="1650190">won't</span> <span m="1650490">go</span> <span m="1650640">through</span> <span m="1650820">all</span> <span m="1650930">the</span> <span m="1651010">details,</span> <span m="1651470">but</span> <span m="1651570">it's</span> <span m="1651690">not</span> <span m="1651890">hard.</span> </p>
<p><span m="1665460">Let's</span> <span m="1666520">see</span> <span m="1666690">if</span> <span m="1666750">I</span> <span m="1666810">got</span> <span m="1667000">that</span> <span m="1667230">right.</span> <span m="1668590">Yeah,</span> <span m="1669060">that</span> <span m="1669140">looks</span> <span m="1669400">right.</span> <span m="1670450">OK,</span> <span m="1670760">so</span> <span m="1670900">that</span> <span m="1671150">is</span> <span m="1671460">the</span> <span m="1671540">closed</span> <span m="1672000">form</span> <span m="1672230">expression</span> <span m="1672780">for</span> <span m="1672870">that</span> <span m="1673070">sum,</span> <span m="1674540">which</span> <span m="1674750">we</span> <span m="1674840">can</span> <span m="1675000">get</span> <span m="1675180">from</span> <span m="1675310">the</span> <span m="1675390">perturbation</span> <span m="1676010">method</span> <span m="1676380">and</span> <span m="1676500">the</span> <span m="1676560">fact</span> <span m="1676850">that</span> <span m="1676950">we'd</span> <span m="1677070">already</span> <span m="1677380">done</span> <span m="1677690">the</span> <span m="1677760">geometric</span> <span m="1678270">series.</span> </p>
<p><span m="1680940">There's</span> <span m="1681130">another</span> <span m="1681510">way</span> <span m="1681780">to</span> <span m="1681850">compute</span> <span m="1682270">these</span> <span m="1682450">kinds</span> <span m="1682780">of</span> <span m="1684020">sums,</span> <span m="1685700">which</span> <span m="1685910">I</span> <span m="1685950">want</span> <span m="1686160">to</span> <span m="1686220">show</span> <span m="1686500">you,</span> <span m="1686760">because</span> <span m="1687320">it</span> <span m="1687450">can</span> <span m="1687690">be</span> <span m="1687800">useful.</span> <span m="1696170">So</span> <span m="1696310">we're</span> <span m="1696370">going</span> <span m="1696490">to</span> <span m="1696550">do</span> <span m="1696610">the</span> <span m="1696710">same</span> <span m="1697120">sum</span> <span m="1697880">and</span> <span m="1698330">derive</span> <span m="1698680">the</span> <span m="1698750">formula</span> <span m="1699180">a</span> <span m="1699210">different</span> <span m="1699530">way.</span> <span m="1701180">This</span> <span m="1701440">method</span> <span m="1701770">is</span> <span m="1701900">called</span> <span m="1702200">the</span> <span m="1702280">derivative</span> <span m="1702780">method,</span> <span m="1711720">and</span> <span m="1711880">the</span> <span m="1711980">idea</span> <span m="1712340">is</span> <span m="1712430">to</span> <span m="1712530">start</span> <span m="1713030">with</span> <span m="1713200">a</span> <span m="1713240">geometric</span> <span m="1713820">series</span> <span m="1714260">which</span> <span m="1714440">it's</span> <span m="1714560">close</span> <span m="1714900">to</span> <span m="1715890">and</span> <span m="1716060">then</span> <span m="1716210">take</span> <span m="1716440">a</span> <span m="1716490">derivative.</span> </p>
<p><span m="1719280">So</span> <span m="1719470">for</span> <span m="1719630">x</span> <span m="1720010">not</span> <span m="1720280">equal</span> <span m="1720490">to</span> <span m="1720570">1,</span> <span m="1721060">we</span> <span m="1721180">already</span> <span m="1721450">know</span> <span m="1722990">from</span> <span m="1723140">the</span> <span m="1723210">theorem</span> <span m="1723700">that</span> <span m="1724210">i</span> <span m="1724520">equals</span> <span m="1724880">0</span> <span m="1725850">to</span> <span m="1726020">n,</span> <span m="1726520">x</span> <span m="1726770">to</span> <span m="1726880">the</span> <span m="1727100">i</span> <span m="1727740">equals</span> <span m="1728690">1</span> <span m="1728950">minus</span> <span m="1729490">x</span> <span m="1729740">to</span> <span m="1729820">the</span> <span m="1729920">n</span> <span m="1730080">plus</span> <span m="1730320">1</span> <span m="1731180">over</span> <span m="1731410">1</span> <span m="1731620">minus</span> <span m="1731970">x.</span> <span m="1733170">That</span> <span m="1733400">was</span> <span m="1733530">the</span> <span m="1733600">theorem.</span> <span m="1734120">We</span> <span m="1734270">already</span> <span m="1734480">know</span> <span m="1734670">that.</span> <span m="1737010">Now,</span> <span m="1737320">I</span> <span m="1737420">can</span> <span m="1737600">take</span> <span m="1737800">the</span> <span m="1737870">derivative</span> <span m="1738520">of</span> <span m="1738630">both</span> <span m="1738920">sides,</span> <span m="1739670">and</span> <span m="1739800">let's</span> <span m="1739950">see</span> <span m="1740140">what</span> <span m="1740270">we</span> <span m="1740360">get</span> <span m="1741040">by</span> <span m="1741150">taking</span> <span m="1741460">the</span> <span m="1741540">derivative.</span> <span m="1743220">Well,</span> <span m="1743410">here</span> <span m="1746370">I</span> <span m="1746510">get</span> <span m="1746660">the</span> <span m="1746730">sum.</span> <span m="1747440">The</span> <span m="1747500">derivative</span> <span m="1747840">of</span> <span m="1747980">x</span> <span m="1748150">to</span> <span m="1748230">the</span> <span m="1748350">i</span> <span m="1748700">is</span> <span m="1748870">just</span> <span m="1748940">i</span> <span m="1749330">times</span> <span m="1749730">x</span> <span m="1749860">to</span> <span m="1749990">the</span> <span m="1750120">i</span> <span m="1750320">minus</span> <span m="1750580">1.</span> </p>
<p><span m="1753020">The</span> <span m="1753170">derivative</span> <span m="1753690">over</span> <span m="1753880">here</span> <span m="1754320">is</span> <span m="1754420">a</span> <span m="1754480">little</span> <span m="1754710">messier.</span> <span m="1756310">I've</span> <span m="1756430">got</span> <span m="1756600">to</span> <span m="1756690">have--</span> <span m="1757480">well,</span> <span m="1757580">I</span> <span m="1757680">take</span> <span m="1758520">1</span> <span m="1758870">minus</span> <span m="1759210">x</span> <span m="1759400">times</span> <span m="1759610">a</span> <span m="1759670">derivative</span> <span m="1760020">of</span> <span m="1760110">that.</span> <span m="1763530">Derivative</span> <span m="1763780">of</span> <span m="1763950">this</span> <span m="1764350">is</span> <span m="1765740">now</span> <span m="1766060">minus</span> <span m="1767480">n</span> <span m="1767750">plus</span> <span m="1768080">1,</span> <span m="1769920">x</span> <span m="1770290">to</span> <span m="1770420">the</span> <span m="1770830">n.</span> <span m="1773020">Then</span> <span m="1773190">I</span> <span m="1773260">take</span> <span m="1774160">this</span> <span m="1774460">times</span> <span m="1774700">the</span> <span m="1774770">derivative</span> <span m="1775190">of</span> <span m="1775425">that</span> <span m="1776980">is</span> <span m="1777220">now</span> <span m="1778055">minus</span> <span m="1778870">minus</span> <span m="1779290">1,</span> <span m="1781160">1</span> <span m="1781420">minus</span> <span m="1781720">x</span> <span m="1781980">to</span> <span m="1782060">the</span> <span m="1782170">n</span> <span m="1782330">plus</span> <span m="1782600">1,</span> <span m="1784790">and</span> <span m="1784910">then</span> <span m="1785040">I</span> <span m="1785130">divide</span> <span m="1785710">by</span> <span m="1787360">that</span> <span m="1787660">squared.</span> </p>
<p><span m="1792730">Now,</span> <span m="1792910">when</span> <span m="1793030">we</span> <span m="1793320">compute</span> <span m="1793710">all</span> <span m="1793900">that</span> <span m="1794130">out,</span> <span m="1794720">we</span> <span m="1794900">get</span> <span m="1795160">this,</span> <span m="1797240">1</span> <span m="1797590">minus</span> <span m="1798150">n</span> <span m="1798390">plus</span> <span m="1798690">1,</span> <span m="1799960">x</span> <span m="1800260">to</span> <span m="1800360">the</span> <span m="1800540">n</span> <span m="1802010">plus</span> <span m="1802400">nx</span> <span m="1803290">to</span> <span m="1803380">the</span> <span m="1803480">n</span> <span m="1803670">plus</span> <span m="1803940">1</span> <span m="1805920">over</span> <span m="1806460">1</span> <span m="1806690">minus</span> <span m="1807070">x</span> <span m="1807310">squared.</span> <span m="1809700">I</span> <span m="1809920">won't</span> <span m="1810180">drag</span> <span m="1810460">you</span> <span m="1810530">through</span> <span m="1810690">the</span> <span m="1810850">algebra,</span> <span m="1811230">but</span> <span m="1811320">it's</span> <span m="1811420">not</span> <span m="1811630">hard</span> <span m="1811830">to</span> <span m="1811900">go</span> <span m="1812070">from</span> <span m="1812240">there</span> <span m="1812400">to</span> <span m="1812470">there.</span> <span m="1813480">This</span> <span m="1813690">is</span> <span m="1813820">pretty</span> <span m="1814040">close</span> <span m="1814390">to</span> <span m="1814470">what</span> <span m="1814600">we</span> <span m="1814710">wanted.</span> <span m="1817020">We're</span> <span m="1817170">trying</span> <span m="1817590">to</span> <span m="1817650">figure</span> <span m="1818020">out</span> <span m="1819420">this,</span> <span m="1821610">and</span> <span m="1821790">we</span> <span m="1821880">almost</span> <span m="1822450">got</span> <span m="1822760">there.</span> <span m="1825540">What</span> <span m="1825700">do</span> <span m="1825790">I</span> <span m="1825880">do</span> <span m="1826440">to</span> <span m="1826560">finish</span> <span m="1826920">it</span> <span m="1827010">up?</span> </p>
<p><span m="1828978">AUDIENCE:</span> <span m="1829101">Multiply</span> <span m="1829224">by</span> <span m="1829347">x.</span> </p>
<p><span m="1829470">PROFESSOR: Multiply</span> <span m="1830060">by</span> <span m="1830280">x.</span> <span m="1830900">Good.</span> <span m="1831730">So</span> <span m="1831970">if</span> <span m="1832070">I</span> <span m="1832150">take</span> <span m="1832430">this</span> <span m="1832700">and</span> <span m="1832790">multiply</span> <span m="1833170">by</span> <span m="1833400">x,</span> <span m="1838450">i</span> <span m="1838705">equals</span> <span m="1838960">zero</span> <span m="1839530">to</span> <span m="1839720">n,</span> <span m="1840140">ix</span> <span m="1841200">to</span> <span m="1841280">the</span> <span m="1841480">i</span> <span m="1841940">equals</span> <span m="1844040">x</span> <span m="1844530">minus</span> <span m="1845180">n</span> <span m="1845430">plus</span> <span m="1845730">1,</span> <span m="1847130">x</span> <span m="1847460">to</span> <span m="1847560">the</span> <span m="1847680">n</span> <span m="1847830">plus</span> <span m="1848140">1,</span> <span m="1849060">plus</span> <span m="1849440">nx</span> <span m="1850200">to</span> <span m="1850290">the</span> <span m="1850410">n</span> <span m="1850550">plus</span> <span m="1850860">2,</span> <span m="1852110">all</span> <span m="1852420">over</span> <span m="1852690">1</span> <span m="1852890">minus</span> <span m="1853230">x</span> <span m="1853400">squared.</span> <span m="1856240">OK?</span> </p>
<p><span m="1857890">Which</span> <span m="1858140">should</span> <span m="1858380">be</span> <span m="1858540">the</span> <span m="1858640">same--</span> <span m="1859120">yeah--</span> <span m="1859270">the</span> <span m="1859360">same</span> <span m="1859590">formula</span> <span m="1859970">we</span> <span m="1860050">had</span> <span m="1860230">up</span> <span m="1860350">there.</span> <span m="1862110">So</span> <span m="1862210">that's</span> <span m="1862310">called</span> <span m="1862520">the</span> <span m="1862590">derivative</span> <span m="1863050">method.</span> <span m="1863400">You</span> <span m="1863470">can</span> <span m="1863570">start</span> <span m="1863840">manipulating--</span> <span m="1864800">you</span> <span m="1864910">treat</span> <span m="1865110">these</span> <span m="1865280">things</span> <span m="1865470">as</span> <span m="1865580">polynomials--</span> <span m="1866490">these</span> <span m="1866570">sums--</span> <span m="1867050">and</span> <span m="1867150">you</span> <span m="1867210">start</span> <span m="1868530">manipulating</span> <span m="1869100">them</span> <span m="1869230">like</span> <span m="1869390">you</span> <span m="1869450">would</span> <span m="1869590">polynomials.</span> <span m="1870930">In</span> <span m="1871035">fact,</span> <span m="1871140">there's</span> <span m="1871280">a</span> <span m="1871350">whole</span> <span m="1871650">branch</span> <span m="1872830">of</span> <span m="1873050">mathematics</span> <span m="1873670">called</span> <span m="1873890">generating</span> <span m="1874460">functions</span> <span m="1874900">that</span> <span m="1875000">we</span> <span m="1875090">won't</span> <span m="1875420">have</span> <span m="1875610">time</span> <span m="1875830">to</span> <span m="1875900">do</span> <span m="1876080">in</span> <span m="1876170">this</span> <span m="1876310">class</span> <span m="1878000">that's</span> <span m="1878200">in</span> <span m="1878340">chapter</span> <span m="1879010">12</span> <span m="1879470">of</span> <span m="1879660">the</span> <span m="1879750">text.</span> <span m="1880560">But</span> <span m="1880690">you</span> <span m="1880900">do</span> <span m="1881110">things</span> <span m="1881500">like</span> <span m="1881750">that</span> <span m="1883170">to</span> <span m="1883690">get</span> <span m="1883980">sums.</span> <span m="1889820">Any</span> <span m="1889990">questions</span> <span m="1890500">about</span> <span m="1891370">what</span> <span m="1891520">we</span> <span m="1891620">did</span> <span m="1891820">there?</span> </p>
<p><span m="1893430">You</span> <span m="1893550">can</span> <span m="1893670">also</span> <span m="1893950">do</span> <span m="1894080">a</span> <span m="1894150">version</span> <span m="1894580">where</span> <span m="1894680">you</span> <span m="1894750">take</span> <span m="1895000">integrals</span> <span m="1896040">of</span> <span m="1896190">this</span> <span m="1896800">if</span> <span m="1896950">you</span> <span m="1897050">want,</span> <span m="1898350">and</span> <span m="1898440">then</span> <span m="1898550">you</span> <span m="1898630">get</span> <span m="1898780">the</span> <span m="1898890">i's</span> <span m="1899260">in</span> <span m="1899350">the</span> <span m="1899420">denominator</span> <span m="1900590">instead</span> <span m="1900870">of</span> <span m="1901100">those</span> <span m="1901240">coefficients.</span> </p>
<p><span m="1905340">For</span> <span m="1905530">homework,</span> <span m="1906400">I</span> <span m="1906550">think</span> <span m="1906810">we've</span> <span m="1906970">given</span> <span m="1907310">you</span> <span m="1908980">the</span> <span m="1909110">sum</span> <span m="1909630">of</span> <span m="1909850">i</span> <span m="1910180">squared</span> <span m="1910870">x</span> <span m="1911100">to</span> <span m="1911190">the</span> <span m="1911330">i.</span> <span m="1912761">How</span> <span m="1913240">do</span> <span m="1913310">you</span> <span m="1913390">think</span> <span m="1913560">you're</span> <span m="1913650">going</span> <span m="1913770">to</span> <span m="1913850">do</span> <span m="1913980">that?</span> <span m="1916640">Any</span> <span m="1916810">thoughts</span> <span m="1917080">about</span> <span m="1917230">how</span> <span m="1917310">you're</span> <span m="1917384">going</span> <span m="1917458">to</span> <span m="1917532">solve</span> <span m="1917606">that?</span> <span m="1919220">Get</span> <span m="1919370">the</span> <span m="1919450">sum,</span> <span m="1919700">a</span> <span m="1919950">closed</span> <span m="1920320">form</span> <span m="1920570">for</span> <span m="1920670">the</span> <span m="1920760">sum</span> <span m="1920990">of</span> <span m="1921180">i</span> <span m="1921440">squared</span> <span m="1921790">x</span> <span m="1921908">to</span> <span m="1922027">the</span> <span m="1922145">i?</span> </p>
<p><span m="1924160">AUDIENCE: Do</span> <span m="1924634">the</span> <span m="1924792">derivative</span> <span m="1924950">method</span> <span m="1925108">twice.</span> </p>
<p><span m="1926060">PROFESSOR:  Yeah,</span> <span m="1926380">do</span> <span m="1926510">it</span> <span m="1926600">twice.</span> <span m="1926920">Take</span> <span m="1927130">this,</span> <span m="1927700">which</span> <span m="1927870">now</span> <span m="1928040">you</span> <span m="1928266">know.</span> <span m="1928946">Take</span> <span m="1929400">the</span> <span m="1929460">derivative</span> <span m="1929820">again.</span> <span m="1931700">Won't</span> <span m="1931790">be</span> <span m="1931930">too</span> <span m="1932710">hard.</span> </p>
<p><span m="1938530">You</span> <span m="1938670">can</span> <span m="1938810">also</span> <span m="1939060">take</span> <span m="1939360">the</span> <span m="1939430">version</span> <span m="1939810">of</span> <span m="1939890">this</span> <span m="1940210">where</span> <span m="1941300">n</span> <span m="1941560">goes</span> <span m="1941810">to</span> <span m="1941920">infinity.</span> <span m="1943980">Let's</span> <span m="1944110">do</span> <span m="1944250">that.</span> <span m="1950120">If</span> <span m="1950560">the</span> <span m="1950670">absolute</span> <span m="1951090">value</span> <span m="1951370">of</span> <span m="1951480">x</span> <span m="1951730">is</span> <span m="1951860">less</span> <span m="1952090">than</span> <span m="1952220">1,</span> <span m="1953700">the</span> <span m="1953800">sum</span> <span m="1954460">i</span> <span m="1954900">equals</span> <span m="1956150">1</span> <span m="1956540">to</span> <span m="1956680">infinity</span> <span m="1958730">of</span> <span m="1959050">ix</span> <span m="1959580">to</span> <span m="1959670">the</span> <span m="1959840">i,</span> <span m="1960870">what</span> <span m="1961050">does</span> <span m="1961240">that</span> <span m="1961530">equal?</span> </p>
<p><span m="1964560">This</span> <span m="1964760">one,</span> <span m="1964880">you</span> <span m="1964950">can</span> <span m="1965060">see</span> <span m="1965320">it</span> <span m="1965380">easier</span> <span m="1965760">up</span> <span m="1965890">here.</span> <span m="1969020">What</span> <span m="1969200">happens</span> <span m="1970700">when</span> <span m="1970970">n</span> <span m="1971310">goes</span> <span m="1971590">to</span> <span m="1971700">infinity?</span> <span m="1973890">What</span> <span m="1974010">does</span> <span m="1974130">this</span> <span m="1974310">do?</span> <span m="1977390">X</span> <span m="1977700">is</span> <span m="1977820">less</span> <span m="1978030">than</span> <span m="1978140">1,</span> <span m="1978650">an</span> <span m="1978780">absolute</span> <span m="1979175">value.</span> <span m="1979570">What</span> <span m="1979800">happens</span> <span m="1980120">to</span> <span m="1980190">this</span> <span m="1980390">as</span> <span m="1980475">n</span> <span m="1980560">goes</span> <span m="1980900">to</span> <span m="1980980">infinity?</span> <span m="1982990">This</span> <span m="1983070">term.</span> <span m="1986380">Goes</span> <span m="1986670">to</span> <span m="1986760">0,</span> <span m="1988650">right?</span> <span m="1989010">This</span> <span m="1989190">gets</span> <span m="1989420">big,</span> <span m="1989780">but</span> <span m="1989900">this</span> <span m="1990060">gets</span> <span m="1990250">smaller</span> <span m="1990800">faster.</span> <span m="1991930">What</span> <span m="1992130">happens</span> <span m="1992450">to</span> <span m="1992520">this</span> <span m="1992720">term</span> <span m="1992960">as</span> <span m="1993090">n</span> <span m="1993230">goes</span> <span m="1993430">to</span> <span m="1993520">infinity?</span> <span m="1996230">Same</span> <span m="1996580">thing,</span> <span m="1997030">0.</span> <span m="1998500">All</span> <span m="1998830">I'm</span> <span m="1998940">left</span> <span m="1999210">with</span> <span m="1999340">is</span> <span m="1999490">x</span> <span m="1999820">over</span> <span m="1999960">1</span> <span m="2000160">minus</span> <span m="2000490">x</span> <span m="2000650">squared.</span> </p>
<p><span m="2009570">Now,</span> <span m="2010180">this</span> <span m="2010510">formula</span> <span m="2011380">is</span> <span m="2011600">useful</span> <span m="2012650">if</span> <span m="2012810">you're</span> <span m="2012930">trying</span> <span m="2013300">to,</span> <span m="2013420">say,</span> <span m="2015260">get</span> <span m="2015410">the</span> <span m="2015490">value</span> <span m="2015870">of</span> <span m="2015950">a</span> <span m="2016020">company,</span> <span m="2017410">and</span> <span m="2017560">the</span> <span m="2017620">company</span> <span m="2018030">is</span> <span m="2018150">growing.</span> <span m="2019490">Every</span> <span m="2019870">year</span> <span m="2020250">the</span> <span m="2020340">company</span> <span m="2021400">grows</span> <span m="2021840">its</span> <span m="2022000">bottom</span> <span m="2022390">line</span> <span m="2022820">by</span> <span m="2023040">m</span> <span m="2023320">dollars.</span> <span m="2024850">So</span> <span m="2025100">the</span> <span m="2025180">first</span> <span m="2025540">year,</span> <span m="2025700">the</span> <span m="2025790">company</span> <span m="2026140">generates</span> <span m="2026580">m</span> <span m="2026780">dollars,</span> <span m="2027240">the</span> <span m="2027310">next</span> <span m="2027610">year</span> <span m="2027770">it</span> <span m="2027860">generates</span> <span m="2028300">two</span> <span m="2028590">m</span> <span m="2028760">dollars</span> <span m="2029220">in</span> <span m="2029320">profit,</span> <span m="2030060">the</span> <span m="2030170">next</span> <span m="2030420">year</span> <span m="2030590">is</span> <span m="2030740">three</span> <span m="2031040">m</span> <span m="2031210">dollars.</span> <span m="2032150">So</span> <span m="2032310">you've</span> <span m="2032410">got</span> <span m="2032600">an</span> <span m="2032710">entity</span> <span m="2033150">that</span> <span m="2033300">every</span> <span m="2033580">year</span> <span m="2033760">is</span> <span m="2033880">growing</span> <span m="2034300">by</span> <span m="2034460">a</span> <span m="2034510">fixed</span> <span m="2034890">amount.</span> <span m="2035260">It's</span> <span m="2035400">not</span> <span m="2035910">doubling</span> <span m="2036440">every</span> <span m="2036640">year,</span> <span m="2037210">but</span> <span m="2037450">every</span> <span m="2037670">year</span> <span m="2037870">adds</span> <span m="2038170">in</span> <span m="2038340">m</span> <span m="2038540">dollars</span> <span m="2038900">more</span> <span m="2039210">of</span> <span m="2039300">profit.</span> </p>
<p><span m="2040760">What</span> <span m="2041090">would</span> <span m="2041240">you</span> <span m="2041380">pay</span> <span m="2041730">to</span> <span m="2041820">buy</span> <span m="2042100">that</span> <span m="2042320">company?</span> <span m="2043400">What</span> <span m="2043620">is</span> <span m="2043720">that</span> <span m="2043940">worth?</span> <span m="2045550">So</span> <span m="2045650">you</span> <span m="2045750">can</span> <span m="2045850">think</span> <span m="2045970">of</span> <span m="2046050">this</span> <span m="2046500">as,</span> <span m="2047450">again,</span> <span m="2047750">an</span> <span m="2047820">annuity.</span> <span m="2053900">Here</span> <span m="2054300">the</span> <span m="2054469">annuity</span> <span m="2057620">pays</span> <span m="2061090">im</span> <span m="2061830">dollars,</span> <span m="2064469">in</span> <span m="2064580">this</span> <span m="2064750">case,</span> <span m="2065090">at</span> <span m="2065179">the</span> <span m="2065400">end,</span> <span m="2067030">not</span> <span m="2067230">the</span> <span m="2067300">beginning,</span> <span m="2067555">say,</span> <span m="2068810">of</span> <span m="2069540">the</span> <span m="2069780">year</span> <span m="2070179">i</span> <span m="2071760">forever.</span> </p>
<p><span m="2079070">This</span> <span m="2079210">company</span> <span m="2079830">is--</span> <span m="2080100">or</span> <span m="2080239">this</span> <span m="2080389">annuity--</span> <span m="2080929">is</span> <span m="2081130">worth,</span> <span m="2083139">well,</span> <span m="2083409">we</span> <span m="2083489">just</span> <span m="2084010">plug</span> <span m="2084330">into</span> <span m="2084489">the</span> <span m="2084570">formula.</span> <span m="2086190">Instead</span> <span m="2086489">of</span> <span m="2087159">$1</span> <span m="2087969">each</span> <span m="2088139">year,</span> <span m="2088300">it's</span> <span m="2088469">m</span> <span m="2088770">so</span> <span m="2088900">there's</span> <span m="2089030">an</span> <span m="2089139">m</span> <span m="2089330">out</span> <span m="2089489">front.</span> <span m="2090469">x</span> <span m="2090790">is</span> <span m="2090929">1</span> <span m="2091130">over</span> <span m="2091260">1</span> <span m="2091460">plus</span> <span m="2091730">p,</span> <span m="2096070">and</span> <span m="2096239">then</span> <span m="2096389">we</span> <span m="2096520">have</span> <span m="2097270">1</span> <span m="2097490">minus</span> <span m="2097930">1</span> <span m="2098120">over</span> <span m="2098280">1</span> <span m="2098490">plus</span> <span m="2098770">p</span> <span m="2099230">squared.</span> <span m="2102620">And</span> <span m="2102680">if</span> <span m="2102760">we</span> <span m="2102850">multiply</span> <span m="2103290">the</span> <span m="2103390">top</span> <span m="2103710">and</span> <span m="2103840">bottom</span> <span m="2104170">by</span> <span m="2104310">1</span> <span m="2104630">plus</span> <span m="2104910">p</span> <span m="2105180">squared,</span> <span m="2106840">we</span> <span m="2107000">get</span> <span m="2108660">m</span> <span m="2109210">1</span> <span m="2109460">plus</span> <span m="2109760">p</span> <span m="2110870">over</span> <span m="2111230">p</span> <span m="2111470">squared.</span> </p>
<p><span m="2115190">So</span> <span m="2115300">it's</span> <span m="2115440">possible</span> <span m="2115715">with</span> <span m="2115990">a</span> <span m="2116020">very</span> <span m="2116320">simple</span> <span m="2116660">formula</span> <span m="2117840">to</span> <span m="2117970">figure</span> <span m="2118270">out</span> <span m="2118520">how</span> <span m="2118710">much</span> <span m="2118920">you</span> <span m="2119010">should</span> <span m="2119190">spend</span> <span m="2119520">to</span> <span m="2119590">buy</span> <span m="2119830">this</span> <span m="2120000">company,</span> <span m="2120930">what</span> <span m="2121130">its</span> <span m="2121260">value</span> <span m="2121680">is</span> <span m="2121820">today.</span> <span m="2122730">So,</span> <span m="2122910">for</span> <span m="2123020">example,</span> <span m="2123500">say</span> <span m="2123760">the</span> <span m="2123840">company</span> <span m="2124230">was</span> <span m="2124910">adding</span> <span m="2125990">$50,000</span> <span m="2127470">a</span> <span m="2127540">year</span> <span m="2129590">in</span> <span m="2129760">profit.</span> <span m="2131650">The</span> <span m="2131840">interest</span> <span m="2132210">rate</span> <span m="2133200">was</span> <span m="2133740">6%,</span> <span m="2135970">the</span> <span m="2136110">value</span> <span m="2136590">of</span> <span m="2136670">this</span> <span m="2136830">company</span> <span m="2137300">is</span> <span m="2137470">$14</span> <span m="2137930">million--</span> <span m="2138190">$14.7</span> <span m="2139560">million,</span> <span m="2144890">just</span> <span m="2145220">plugging</span> <span m="2145510">into</span> <span m="2145680">that</span> <span m="2145860">formula.</span> </p>
<p><span m="2147960">So</span> <span m="2148070">people</span> <span m="2148390">that</span> <span m="2149580">buy</span> <span m="2149900">companies</span> <span m="2150440">and</span> <span m="2150550">stuff,</span> <span m="2151380">they</span> <span m="2151550">use</span> <span m="2151800">formulas</span> <span m="2152410">like</span> <span m="2152630">this</span> <span m="2153000">to</span> <span m="2153580">figure</span> <span m="2154050">out</span> <span m="2155010">what</span> <span m="2155200">it's</span> <span m="2155340">worth.</span> <span m="2155990">Of</span> <span m="2156050">course,</span> <span m="2156390">you've</span> <span m="2156410">got</span> <span m="2156500">to</span> <span m="2156570">make</span> <span m="2156730">sure</span> <span m="2156910">it's</span> <span m="2157090">really</span> <span m="2157320">going</span> <span m="2157440">to</span> <span m="2157500">keep</span> <span m="2157710">paying</span> <span m="2158690">the</span> <span m="2158930">$50,000</span> <span m="2159810">more</span> <span m="2160470">every</span> <span m="2160790">year</span> <span m="2161780">and</span> <span m="2161970">that</span> <span m="2162120">this</span> <span m="2162320">is</span> <span m="2162530">the</span> <span m="2162630">right</span> <span m="2162810">interest</span> <span m="2163160">rate</span> <span m="2163370">to</span> <span m="2163470">be</span> <span m="2163550">thinking</span> <span m="2163870">about.</span> </p>
<p><span m="2165910">You</span> <span m="2165977">know,</span> <span m="2166045">and</span> <span m="2166180">the</span> <span m="2166240">guys</span> <span m="2166540">on</span> <span m="2166660">Wall</span> <span m="2166890">Street--</span> <span m="2167090">the</span> <span m="2167160">bankers</span> <span m="2167530">on</span> <span m="2167650">Wall</span> <span m="2167860">Street--</span> <span m="2168070">they</span> <span m="2168200">all</span> <span m="2168330">have</span> <span m="2169190">their</span> <span m="2169350">estimations</span> <span m="2170040">for</span> <span m="2170160">what</span> <span m="2170350">these</span> <span m="2170600">things</span> <span m="2171650">are--</span> <span m="2172670">the</span> <span m="2172780">value</span> <span m="2173080">of</span> <span m="2173160">p</span> <span m="2173400">they</span> <span m="2173510">would</span> <span m="2173600">put</span> <span m="2173800">into</span> <span m="2173950">these</span> <span m="2174120">formulas.</span> <span m="2176200">Any</span> <span m="2176360">questions</span> <span m="2176780">about</span> <span m="2177010">that?</span> <span m="2179410">Yeah.</span> </p>
<p><span m="2180715">AUDIENCE:</span> <span m="2181150">[INAUDIBLE]</span> </p>
<p><span m="2182460">PROFESSOR: OK.</span> <span m="2182830">Good.</span> <span m="2184290">So</span> <span m="2185340">this</span> <span m="2185600">one</span> <span m="2185770">is</span> <span m="2185880">OK?</span> <span m="2187170">OK.</span> <span m="2188040">I</span> <span m="2188210">plugged</span> <span m="2189350">x</span> <span m="2189850">equals</span> <span m="2190380">1</span> <span m="2190630">over</span> <span m="2190870">1</span> <span m="2191180">plus</span> <span m="2191530">p,</span> <span m="2192910">like</span> <span m="2193110">we</span> <span m="2193200">did</span> <span m="2193370">before--</span> <span m="2194050">remember</span> <span m="2194340">for</span> <span m="2194450">the</span> <span m="2194550">annuity--</span> <span m="2194920">because</span> <span m="2195770">every</span> <span m="2196120">year</span> <span m="2197160">you're</span> <span m="2197350">degrading</span> <span m="2198000">it,</span> <span m="2198090">devaluing</span> <span m="2198750">by</span> <span m="2198870">1</span> <span m="2199200">over</span> <span m="2199380">1</span> <span m="2199550">plus</span> <span m="2199770">p.</span> <span m="2200430">So</span> <span m="2200580">that's</span> <span m="2200800">the</span> <span m="2200960">x</span> <span m="2201260">term,</span> <span m="2202720">and</span> <span m="2203680">it's</span> <span m="2203940">paying--</span> <span m="2205300">in</span> <span m="2205750">the</span> <span m="2206030">i-th</span> <span m="2206340">year,</span> <span m="2207160">it's</span> <span m="2207460">paying</span> <span m="2208620">im</span> <span m="2209310">dollars.</span> <span m="2211400">All</span> <span m="2211570">right?</span> </p>
<p><span m="2212250">So</span> <span m="2212480">the</span> <span m="2212560">first</span> <span m="2212980">year</span> <span m="2213190">it</span> <span m="2213280">pays</span> <span m="2213670">n</span> <span m="2213890">dollars,</span> <span m="2214940">the</span> <span m="2215060">next</span> <span m="2215360">year</span> <span m="2215610">2</span> <span m="2215840">m,</span> <span m="2216470">the</span> <span m="2216560">next</span> <span m="2216850">year</span> <span m="2216990">3</span> <span m="2217420">m,</span> <span m="2218270">the</span> <span m="2218340">next</span> <span m="2218580">year</span> <span m="2218700">4</span> <span m="2219120">m,</span> <span m="2220830">but</span> <span m="2221110">every</span> <span m="2221400">year</span> <span m="2221590">you're</span> <span m="2221720">knocking</span> <span m="2222100">it</span> <span m="2222200">down</span> <span m="2222490">by</span> <span m="2222610">1</span> <span m="2222910">plus</span> <span m="2223140">1</span> <span m="2223310">over</span> <span m="2223500">p</span> <span m="2223940">to</span> <span m="2224110">the</span> <span m="2224250">number</span> <span m="2224510">of</span> <span m="2224600">years.</span> <span m="2226070">So</span> <span m="2226290">what</span> <span m="2226470">you</span> <span m="2226570">get--</span> <span m="2226900">the</span> <span m="2226980">sum</span> <span m="2227260">you've</span> <span m="2227400">really</span> <span m="2227620">got</span> <span m="2227890">here--</span> <span m="2231520">is</span> <span m="2231690">i</span> <span m="2231880">equals</span> <span m="2232020">1</span> <span m="2232180">to</span> <span m="2232290">infinity,</span> <span m="2233250">im</span> <span m="2233740">dollars</span> <span m="2234180">are</span> <span m="2234250">paid,</span> <span m="2235130">but</span> <span m="2235310">those</span> <span m="2235540">dollars</span> <span m="2236030">are</span> <span m="2236110">worth</span> <span m="2237320">1</span> <span m="2237550">over</span> <span m="2237690">1</span> <span m="2237860">plus</span> <span m="2238120">p</span> <span m="2238300">to</span> <span m="2238390">the</span> <span m="2238550">i</span> <span m="2239700">today,</span> <span m="2240280">the</span> <span m="2240380">current</span> <span m="2240680">value.</span> <span m="2241370">That's</span> <span m="2241590">a</span> <span m="2241630">good</span> <span m="2241770">question.</span> <span m="2242110">I</span> <span m="2242160">should've</span> <span m="2243080">said</span> <span m="2243300">that.</span> <span m="2244350">That's</span> <span m="2244520">a</span> <span m="2244570">great</span> <span m="2244780">question.</span> </p>
<p><span m="2246670">So</span> <span m="2246720">that's</span> <span m="2247030">how</span> <span m="2247150">we</span> <span m="2247250">connected</span> <span m="2247690">this</span> <span m="2247870">up,</span> <span m="2248030">because</span> <span m="2248180">you're</span> <span m="2248280">getting</span> <span m="2248570">paid</span> <span m="2248920">this</span> <span m="2249140">much</span> <span m="2249830">in</span> <span m="2250010">our</span> <span m="2250180">years,</span> <span m="2251130">and</span> <span m="2251320">that's</span> <span m="2251490">worth</span> <span m="2252160">that</span> <span m="2252470">much</span> <span m="2252690">degradation</span> <span m="2253390">or</span> <span m="2253470">that</span> <span m="2253620">much</span> <span m="2253800">devaluation</span> <span m="2254520">today,</span> <span m="2255130">and</span> <span m="2255270">now</span> <span m="2255540">we</span> <span m="2255740">add</span> <span m="2255780">up</span> <span m="2255820">a</span> <span m="2255870">total</span> <span m="2256140">current</span> <span m="2256420">value.</span> <span m="2258930">So</span> <span m="2259050">even</span> <span m="2259300">a</span> <span m="2259340">company</span> <span m="2259760">that</span> <span m="2259860">is</span> <span m="2259970">paying</span> <span m="2260240">you</span> <span m="2260330">more</span> <span m="2260710">and</span> <span m="2260870">more</span> <span m="2261190">every</span> <span m="2261510">year</span> <span m="2261980">still</span> <span m="2262200">has</span> <span m="2262370">a</span> <span m="2262430">finite</span> <span m="2262820">value,</span> <span m="2264740">because</span> <span m="2266030">the</span> <span m="2266240">extra--</span> <span m="2267220">the</span> <span m="2267300">payments</span> <span m="2267900">are</span> <span m="2268020">increasing</span> <span m="2268570">but</span> <span m="2268670">only</span> <span m="2268890">linearly.</span> <span m="2269990">The</span> <span m="2270090">value</span> <span m="2270480">today</span> <span m="2270770">is</span> <span m="2270870">decreasing</span> <span m="2271360">geometrically,</span> <span m="2272670">and</span> <span m="2272820">the</span> <span m="2272890">geometric</span> <span m="2273490">decrease</span> <span m="2273960">wipes</span> <span m="2274390">out</span> <span m="2274630">the</span> <span m="2274700">value</span> <span m="2275040">of</span> <span m="2275100">the</span> <span m="2275160">company</span> <span m="2275480">in</span> <span m="2275540">the</span> <span m="2275600">future.</span> <span m="2276600">Yeah.</span> </p>
<p><span m="2277100">AUDIENCE:</span> <span m="2277600">Are you</span> <span m="2278600">[? squaring ?]</span> <span m="2280600">quantity</span> <span m="2281003">of</span> <span m="2281406">[INAUDIBLE].</span> </p>
<p><span m="2285640">PROFESSOR: What</span> <span m="2285750">did</span> <span m="2285850">I</span> <span m="2285920">do?</span> <span m="2286780">Oh,</span> <span m="2287040">wait,</span> <span m="2287350">wait,</span> <span m="2287740">wait.</span> <span m="2289030">I</span> <span m="2289170">screwed</span> <span m="2289440">up</span> <span m="2289610">here</span> <span m="2290071">too.</span> <span m="2292380">Is</span> <span m="2292510">that</span> <span m="2292660">what</span> <span m="2292760">you're</span> <span m="2292870">asking</span> <span m="2293110">about?</span> <span m="2294700">Yeah.</span> <span m="2295160">That's</span> <span m="2295390">what</span> <span m="2295490">I</span> <span m="2295530">should</span> <span m="2295680">have</span> <span m="2295770">done,</span> <span m="2295970">right?</span> <span m="2296140">Because</span> <span m="2296290">I</span> <span m="2296340">got</span> <span m="2296550">1</span> <span m="2297020">minus</span> <span m="2297450">x</span> <span m="2298250">is</span> <span m="2298430">1</span> <span m="2298730">minus</span> <span m="2299050">1</span> <span m="2299270">over</span> <span m="2299340">1</span> <span m="2299580">plus</span> <span m="2299830">p.</span> <span m="2300600">That</span> <span m="2301200">gets</span> <span m="2301400">squared,</span> <span m="2302800">and</span> <span m="2302960">now</span> <span m="2303260">when</span> <span m="2303380">I</span> <span m="2303420">multiply</span> <span m="2303980">1</span> <span m="2304250">plus</span> <span m="2304510">p</span> <span m="2304740">squared,</span> <span m="2305970">it's</span> <span m="2306150">multiplying</span> <span m="2306570">this</span> <span m="2306740">by</span> <span m="2306840">1</span> <span m="2307040">plus</span> <span m="2307310">p.</span> <span m="2308310">It's</span> <span m="2308610">1</span> <span m="2308850">plus</span> <span m="2309080">p</span> <span m="2309280">minus</span> <span m="2309730">1</span> <span m="2310020">is</span> <span m="2310170">p.</span> <span m="2310630">So</span> <span m="2310770">I</span> <span m="2310800">have</span> <span m="2310910">p</span> <span m="2311070">squared.</span> <span m="2312490">All</span> <span m="2312640">right,</span> <span m="2312780">so</span> <span m="2312855">this</span> <span m="2312930">part's</span> <span m="2313200">OK.</span> <span m="2313550">That</span> <span m="2313780">part</span> <span m="2313980">I</span> <span m="2314050">wrote</span> <span m="2314270">wrong.</span> <span m="2315610">That's</span> <span m="2315890">good.</span> <span m="2316280">Any</span> <span m="2316480">other</span> <span m="2316630">questions?</span> </p>
<p><span m="2326430">So</span> <span m="2326540">let's</span> <span m="2326680">do</span> <span m="2326820">a</span> <span m="2326880">simple</span> <span m="2327280">example.</span> <span m="2343230">What</span> <span m="2343550">is</span> <span m="2344380">this</span> <span m="2344610">sum?</span> <span m="2354026">A</span> <span m="2354510">1/2</span> <span m="2355270">plus</span> <span m="2356480">2/4</span> <span m="2357910">plus</span> <span m="2358310">3/8</span> <span m="2359630">plus</span> <span m="2360180">4/16</span> <span m="2363060">forever.</span> <span m="2363580">What's</span> <span m="2363930">that</span> <span m="2364700">sum</span> <span m="2366130">equal?</span> <span m="2372970">You</span> <span m="2373120">can</span> <span m="2373270">plug</span> <span m="2373420">that</span> <span m="2373580">in</span> <span m="2373640">the</span> <span m="2373700">formula.</span> </p>
<p><span m="2375975">AUDIENCE:</span> <span m="2376430">[INAUDIBLE]</span> </p>
<p><span m="2378060">PROFESSOR: Yeah.</span> <span m="2380440">That's</span> <span m="2380650">right.</span> <span m="2381260">Good.</span> <span m="2383350">These</span> <span m="2383530">details.</span> <span m="2386310">Otherwise</span> <span m="2386710">that</span> <span m="2386890">sum</span> <span m="2387140">would</span> <span m="2387260">be</span> <span m="2387370">what?</span> <span m="2388156">If</span> <span m="2388550">I</span> <span m="2388660">didn't</span> <span m="2388810">put</span> <span m="2388930">the</span> <span m="2389000">negative</span> <span m="2389430">here,</span> <span m="2390640">it's</span> <span m="2390730">going</span> <span m="2390840">to</span> <span m="2391030">be</span> <span m="2391400">infinity,</span> <span m="2391500">and</span> <span m="2391700">that's</span> <span m="2391900">not</span> <span m="2392150">so</span> <span m="2392360">interesting.</span> <span m="2393510">The</span> <span m="2393640">negative</span> <span m="2394030">makes</span> <span m="2394270">it</span> <span m="2394340">more</span> <span m="2394540">interesting,</span> <span m="2394970">so</span> <span m="2395120">I</span> <span m="2395170">got</span> <span m="2395380">1</span> <span m="2395580">over</span> <span m="2395740">2</span> <span m="2395970">to</span> <span m="2396035">the</span> <span m="2396100">i</span> <span m="2396280">in</span> <span m="2396400">there.</span> <span m="2397810">What's</span> <span m="2398030">it</span> <span m="2398110">worth</span> <span m="2398340">then?</span> <span m="2401790">Well,</span> <span m="2402180">I</span> <span m="2402240">can</span> <span m="2402350">plug</span> <span m="2402545">in</span> <span m="2402740">the</span> <span m="2402810">formula.</span> <span m="2404030">What's</span> <span m="2404310">x?</span> <span m="2406650">One</span> <span m="2407060">half.</span> <span m="2407760">So</span> <span m="2407940">I</span> <span m="2408010">get</span> <span m="2408170">1/2</span> <span m="2409150">over</span> <span m="2409560">1</span> <span m="2409840">minus</span> <span m="2410250">1/2</span> <span m="2410870">squared</span> <span m="2412670">is</span> <span m="2413500">1/2</span> <span m="2414510">over</span> <span m="2415300">1/4,</span> <span m="2416740">and</span> <span m="2416910">that's</span> <span m="2417270">2.</span> <span m="2424710">Any</span> <span m="2424860">questions</span> <span m="2425300">on</span> <span m="2425420">that</span> <span m="2426250">formula?</span> <span m="2428925">It's</span> <span m="2429210">amazing</span> <span m="2429550">how</span> <span m="2429700">useful</span> <span m="2430080">these</span> <span m="2430270">things</span> <span m="2430470">get</span> <span m="2430620">to</span> <span m="2430690">be</span> <span m="2430800">later.</span> </p>
<p><span m="2434910">So</span> <span m="2435230">that's</span> <span m="2435710">sort</span> <span m="2435870">of</span> <span m="2435940">the</span> <span m="2436000">geometric</span> <span m="2436940">kinds</span> <span m="2437400">of</span> <span m="2437490">things.</span> <span m="2439130">Next</span> <span m="2439420">I</span> <span m="2439460">want</span> <span m="2439630">to</span> <span m="2439700">talk</span> <span m="2439980">about</span> <span m="2440710">more</span> <span m="2440940">of</span> <span m="2441000">the</span> <span m="2441120">arithmetic</span> <span m="2441910">kinds</span> <span m="2442260">of</span> <span m="2442340">sums</span> <span m="2443160">and</span> <span m="2443310">what</span> <span m="2443480">you</span> <span m="2443550">do</span> <span m="2443720">there.</span> <span m="2445440">In</span> <span m="2445600">fact,</span> <span m="2445930">we've</span> <span m="2445970">already</span> <span m="2446200">seen</span> <span m="2446680">one</span> <span m="2447020">that</span> <span m="2447110">we've</span> <span m="2447270">done.</span> <span m="2449210">If</span> <span m="2449390">I</span> <span m="2449550">sum</span> <span m="2450890">i</span> <span m="2451230">equals</span> <span m="2451680">1</span> <span m="2452050">to</span> <span m="2452160">n</span> <span m="2452960">of</span> <span m="2453120">i--</span> <span m="2454110">I</span> <span m="2454570">think</span> <span m="2454730">we've</span> <span m="2454900">already</span> <span m="2455100">done</span> <span m="2455390">this</span> <span m="2455610">one--</span> <span m="2455860">that's</span> <span m="2456080">just</span> <span m="2456370">n</span> <span m="2457310">times</span> <span m="2457420">n</span> <span m="2457700">plus</span> <span m="2457990">1</span> <span m="2458460">over</span> <span m="2458720">2,</span> <span m="2459500">and</span> <span m="2459890">probably</span> <span m="2460220">most</span> <span m="2460530">of</span> <span m="2460610">you</span> <span m="2460740">even</span> <span m="2460870">learned</span> <span m="2461170">that</span> <span m="2461390">formula</span> <span m="2461820">back</span> <span m="2462650">in</span> <span m="2462810">middle</span> <span m="2463050">school,</span> <span m="2463380">I'm</span> <span m="2463500">guessing--</span> <span m="2464170">maybe</span> <span m="2464440">before.</span> </p>
<p><span m="2467220">How</span> <span m="2467410">many</span> <span m="2467620">people</span> <span m="2468210">know</span> <span m="2468460">the</span> <span m="2468630">answer</span> <span m="2468950">for</span> <span m="2469090">this</span> <span m="2469370">sum?</span> <span m="2475000">The</span> <span m="2475110">sum</span> <span m="2475520">of</span> <span m="2475650">the</span> <span m="2475720">squares--</span> <span m="2476650">the</span> <span m="2476750">first</span> <span m="2476990">n</span> <span m="2477130">squares.</span> <span m="2477970">Somebody</span> <span m="2478210">knows.</span> <span m="2478390">What</span> <span m="2478770">is</span> <span m="2478870">it?</span> </p>
<p><span m="2479720">AUDIENCE:</span> <span m="2480080">n</span> <span m="2480850">times</span> <span m="2481240">n</span> <span m="2481430">plus</span> <span m="2481650">1</span> <span m="2483378">times</span> <span m="2483594">2n</span> <span m="2483940">plus</span> <span m="2484180">1,</span> <span m="2485590">all</span> <span m="2485770">over</span> <span m="2485950">6.</span> </p>
<p><span m="2486940">PROFESSOR: Very</span> <span m="2487220">good.</span> <span m="2487470">That</span> <span m="2487630">is</span> <span m="2487800">correct.</span> <span m="2489160">Most</span> <span m="2489410">people</span> <span m="2489650">don't</span> <span m="2490610">remember</span> <span m="2491020">that</span> <span m="2491230">one.</span> <span m="2491450">It's</span> <span m="2491485">a</span> <span m="2491520">little</span> <span m="2491790">harder</span> <span m="2492250">to</span> <span m="2492420">derive.</span> <span m="2495140">How</span> <span m="2495370">would</span> <span m="2495490">you</span> <span m="2495570">prove</span> <span m="2495940">this</span> <span m="2497980">by</span> <span m="2498230">induction?</span> <span m="2499410">Unfortunately,</span> <span m="2500080">induction</span> <span m="2500550">doesn't</span> <span m="2500860">tell</span> <span m="2501060">you</span> <span m="2501200">how</span> <span m="2501380">to</span> <span m="2501490">remember</span> <span m="2502000">what</span> <span m="2502200">the</span> <span m="2502270">formula</span> <span m="2502720">was,</span> <span m="2504260">and</span> <span m="2504480">there's</span> <span m="2504630">a</span> <span m="2504680">couple</span> <span m="2504980">of</span> <span m="2505060">ways</span> <span m="2505410">you</span> <span m="2505520">can</span> <span m="2506050">go</span> <span m="2506220">about</span> <span m="2506540">that.</span> </p>
<p><span m="2507700">One</span> <span m="2508070">is,</span> <span m="2508330">you</span> <span m="2508450">can</span> <span m="2508860">remember</span> <span m="2509580">or</span> <span m="2509780">guess</span> <span m="2510460">that</span> <span m="2510640">the</span> <span m="2510830">answer</span> <span m="2511240">is</span> <span m="2511410">a</span> <span m="2511500">polynomial</span> <span m="2511885">in</span> <span m="2512270">n.</span> <span m="2513222">In</span> <span m="2513700">fact,</span> <span m="2514080">because</span> <span m="2514720">you're</span> <span m="2514860">summing</span> <span m="2515180">squares,</span> <span m="2515710">you</span> <span m="2515800">might</span> <span m="2516040">guess</span> <span m="2516520">that</span> <span m="2517380">it's</span> <span m="2517680">a</span> <span m="2517770">cubic</span> <span m="2518260">polynomial</span> <span m="2518880">in</span> <span m="2518940">n,</span> <span m="2520100">and</span> <span m="2520270">if</span> <span m="2520360">you</span> <span m="2520480">remember</span> <span m="2520840">just</span> <span m="2521140">that</span> <span m="2521380">or</span> <span m="2521430">guess</span> <span m="2521780">just</span> <span m="2522130">that,</span> <span m="2523340">then</span> <span m="2523800">you</span> <span m="2523890">could</span> <span m="2524030">actually</span> <span m="2524340">plug</span> <span m="2524555">in</span> <span m="2524770">values</span> <span m="2525380">and</span> <span m="2525490">get</span> <span m="2525640">the</span> <span m="2525770">answer.</span> </p>
<p><span m="2528540">And</span> <span m="2528680">this</span> <span m="2528870">is--</span> <span m="2529370">you</span> <span m="2529585">know,</span> <span m="2529800">a</span> <span m="2530160">common</span> <span m="2530570">method</span> <span m="2530870">of</span> <span m="2530980">solving</span> <span m="2531410">these</span> <span m="2531600">sums</span> <span m="2531960">is</span> <span m="2532070">you</span> <span m="2532160">sort</span> <span m="2532360">of</span> <span m="2532450">guess</span> <span m="2532850">the</span> <span m="2532920">form</span> <span m="2533290">of</span> <span m="2533370">the</span> <span m="2533440">solution.</span> <span m="2535010">In</span> <span m="2535150">this</span> <span m="2535330">case</span> <span m="2537150">you</span> <span m="2537310">might</span> <span m="2537640">guess</span> <span m="2542070">that</span> <span m="2542280">for</span> <span m="2542470">all</span> <span m="2542760">n,</span> <span m="2543760">the</span> <span m="2543940">sum</span> <span m="2544480">i</span> <span m="2544810">equals</span> <span m="2545180">1</span> <span m="2545480">to</span> <span m="2545610">n</span> <span m="2545920">of</span> <span m="2546015">i</span> <span m="2546110">squared</span> <span m="2547570">equals</span> <span m="2548710">a</span> <span m="2548830">cubic.</span> <span m="2554730">And</span> <span m="2555310">then</span> <span m="2555560">what</span> <span m="2555750">you</span> <span m="2555850">would</span> <span m="2555970">do</span> <span m="2556300">is</span> <span m="2556460">plug</span> <span m="2556640">in</span> <span m="2558300">the</span> <span m="2558400">value</span> <span m="2559160">n</span> <span m="2559360">equals</span> <span m="2560710">1,</span> <span m="2561380">n</span> <span m="2561560">equals</span> <span m="2562110">2,</span> <span m="2562490">maybe</span> <span m="2562680">even--</span> <span m="2563520">we'll</span> <span m="2563650">make</span> <span m="2563860">it</span> <span m="2564040">n</span> <span m="2564200">equals</span> <span m="2564520">0--</span> <span m="2565510">make</span> <span m="2565690">it</span> <span m="2565790">simple</span> <span m="2566940">and</span> <span m="2567080">start</span> <span m="2567350">getting</span> <span m="2567560">some</span> <span m="2568120">constraints</span> <span m="2568770">on</span> <span m="2568970">the</span> <span m="2569380">coefficients.</span> <span m="2571380">If</span> <span m="2571480">you</span> <span m="2571610">would</span> <span m="2571730">plug</span> <span m="2571968">in</span> <span m="2575480">n</span> <span m="2575940">equals</span> <span m="2576730">0,</span> <span m="2577990">the</span> <span m="2578110">sum</span> <span m="2578560">is</span> <span m="2578680">0.</span> </p>
<p><span m="2582300">The</span> <span m="2582400">polynomial</span> <span m="2583120">evaluates</span> <span m="2583810">to</span> <span m="2583920">d.</span> <span m="2586250">That</span> <span m="2586360">tells</span> <span m="2586570">you</span> <span m="2586670">what</span> <span m="2586790">d</span> <span m="2586920">has</span> <span m="2587010">got</span> <span m="2587140">to</span> <span m="2587200">be</span> <span m="2587300">right</span> <span m="2587470">away.</span> <span m="2587935">n</span> <span m="2588400">equals</span> <span m="2588710">1.</span> <span m="2590160">The</span> <span m="2590290">sum</span> <span m="2590800">is</span> <span m="2591010">1,</span> <span m="2592520">and</span> <span m="2592760">when</span> <span m="2592870">you</span> <span m="2592940">plug</span> <span m="2593240">into</span> <span m="2593400">the</span> <span m="2593480">polynomial,</span> <span m="2594050">you</span> <span m="2594110">get</span> <span m="2594260">a</span> <span m="2594720">plus</span> <span m="2595020">b</span> <span m="2595890">plus</span> <span m="2596170">c</span> <span m="2596560">plus</span> <span m="2596860">d.</span> <span m="2598760">n</span> <span m="2598790">equals</span> <span m="2599110">2.</span> <span m="2601540">Well,</span> <span m="2601940">that's</span> <span m="2602200">1</span> <span m="2602440">plus</span> <span m="2602660">4</span> <span m="2602960">is</span> <span m="2603070">5,</span> <span m="2605540">2</span> <span m="2605840">cubed</span> <span m="2606230">is</span> <span m="2606430">8,</span> <span m="2606710">so</span> <span m="2606855">you</span> <span m="2607000">have</span> <span m="2607145">8a</span> <span m="2608450">plus</span> <span m="2608710">4b</span> <span m="2610160">plus</span> <span m="2610440">2c</span> <span m="2611170">plus</span> <span m="2611380">d,</span> <span m="2612680">and</span> <span m="2612930">you'll</span> <span m="2613030">need</span> <span m="2613260">one</span> <span m="2613540">more</span> <span m="2613860">since</span> <span m="2614070">you've</span> <span m="2614170">got</span> <span m="2614340">four</span> <span m="2614600">variables.</span> <span m="2616720">Let's</span> <span m="2616920">see,</span> <span m="2617105">1</span> <span m="2617290">plus</span> <span m="2617570">4</span> <span m="2617870">plus</span> <span m="2618130">9</span> <span m="2618420">is</span> <span m="2618530">14.</span> <span m="2620950">I've</span> <span m="2621300">now</span> <span m="2621590">got</span> <span m="2622160">3</span> <span m="2622490">cubed</span> <span m="2622980">is</span> <span m="2623385">27a</span> <span m="2625100">plus</span> <span m="2626190">9b</span> <span m="2627270">plus</span> <span m="2627640">3c</span> <span m="2628510">plus</span> <span m="2628760">d.</span> </p>
<p><span m="2630690">So</span> <span m="2630850">now</span> <span m="2631040">I've</span> <span m="2631130">got</span> <span m="2631320">four</span> <span m="2631640">equations</span> <span m="2632230">and</span> <span m="2632350">four</span> <span m="2632590">variables</span> <span m="2634120">and,</span> <span m="2634580">with</span> <span m="2634750">any</span> <span m="2634940">luck,</span> <span m="2635240">I</span> <span m="2635280">can</span> <span m="2635420">solve</span> <span m="2635730">that</span> <span m="2635900">system</span> <span m="2636250">of</span> <span m="2636330">equations</span> <span m="2637360">and</span> <span m="2637510">get</span> <span m="2637660">the</span> <span m="2637780">answer.</span> <span m="2638930">And,</span> <span m="2639120">in</span> <span m="2639180">fact,</span> <span m="2639490">you</span> <span m="2639570">can.</span> <span m="2640090">When</span> <span m="2640240">you</span> <span m="2640310">solve</span> <span m="2640690">this</span> <span m="2640780">system,</span> <span m="2641290">you</span> <span m="2641390">get</span> <span m="2642790">a</span> <span m="2643120">equals</span> <span m="2643470">1/3,</span> <span m="2645100">b</span> <span m="2645520">equals</span> <span m="2645870">1/2,</span> <span m="2647300">c</span> <span m="2647790">equals</span> <span m="2648110">1/6,</span> <span m="2649250">and</span> <span m="2649500">d</span> <span m="2649710">equals</span> <span m="2649920">0.</span> <span m="2651460">And</span> <span m="2651640">that's</span> <span m="2651830">exactly</span> <span m="2652340">what</span> <span m="2652510">you</span> <span m="2652600">get</span> <span m="2653300">in</span> <span m="2653390">that</span> <span m="2653540">formula.</span> <span m="2654970">So</span> <span m="2655000">that's</span> <span m="2655150">a</span> <span m="2655200">way</span> <span m="2655350">to</span> <span m="2655460">reproduce</span> <span m="2656240">the</span> <span m="2656320">formula</span> <span m="2656750">if</span> <span m="2656840">you</span> <span m="2656930">forgot</span> <span m="2657310">it.</span> </p>
<p><span m="2659650">Now,</span> <span m="2660710">this</span> <span m="2661070">method--</span> <span m="2662580">really</span> <span m="2662900">to</span> <span m="2662960">be</span> <span m="2663090">sure</span> <span m="2663330">you</span> <span m="2663410">got</span> <span m="2663560">the</span> <span m="2663650">right</span> <span m="2663850">answer--</span> <span m="2665200">you've</span> <span m="2665350">got</span> <span m="2665460">to</span> <span m="2665520">go</span> <span m="2665620">prove</span> <span m="2665830">it</span> <span m="2665920">by</span> <span m="2666080">induction,</span> <span m="2667490">because</span> <span m="2667650">I</span> <span m="2668960">derived</span> <span m="2669810">the</span> <span m="2669980">answer--</span> <span m="2670510">if</span> <span m="2670720">it</span> <span m="2670800">was</span> <span m="2670920">a</span> <span m="2670990">polynomial,</span> <span m="2671740">I</span> <span m="2671840">would</span> <span m="2671960">have</span> <span m="2672050">gotten</span> <span m="2672230">it</span> <span m="2672340">right,</span> <span m="2672530">but</span> <span m="2672640">I</span> <span m="2672690">might</span> <span m="2672900">be</span> <span m="2673000">wrong</span> <span m="2673250">in</span> <span m="2673320">my</span> <span m="2673420">guess.</span> <span m="2674480">And</span> <span m="2674580">to</span> <span m="2674640">make</span> <span m="2674790">sure</span> <span m="2674900">your</span> <span m="2675020">guess</span> <span m="2675330">is</span> <span m="2675440">right,</span> <span m="2675710">you've</span> <span m="2675800">got</span> <span m="2675910">to</span> <span m="2675970">go</span> <span m="2676070">back</span> <span m="2676330">and</span> <span m="2676430">use</span> <span m="2676590">induction</span> <span m="2676990">to</span> <span m="2677070">prove</span> <span m="2677300">it</span> <span m="2678060">for</span> <span m="2678170">this</span> <span m="2678310">approach.</span> <span m="2678786">Yeah.</span> </p>
<p><span m="2679262">AUDIENCE:</span> <span m="2679740">How do you know</span> <span m="2680143">that it would be</span> <span m="2680546">[INAUDIBLE]</span> <span m="2681392">and</span> <span m="2681834">not</span> <span m="2681981">some</span> <span m="2682128">higher</span> <span m="2682276">power?</span> </p>
<p><span m="2683160">PROFESSOR: Well,</span> <span m="2683630">it</span> <span m="2683800">turns</span> <span m="2683970">out</span> <span m="2684680">that</span> <span m="2685040">anytime</span> <span m="2685580">you're</span> <span m="2685710">summing</span> <span m="2686520">powers,</span> <span m="2687570">the</span> <span m="2687720">answer</span> <span m="2688130">is</span> <span m="2688640">a</span> <span m="2688790">polynomial</span> <span m="2689380">to</span> <span m="2689460">one</span> <span m="2689630">higher</span> <span m="2689840">degree.</span> <span m="2691540">So</span> <span m="2691900">if</span> <span m="2692150">you</span> <span m="2692210">just</span> <span m="2692380">remembered</span> <span m="2692640">that</span> <span m="2692860">fact,</span> <span m="2693065">or</span> <span m="2693270">you</span> <span m="2693380">guessed</span> <span m="2693720">that</span> <span m="2693920">fact.</span> <span m="2694750">Another</span> <span m="2695040">way</span> <span m="2695160">to</span> <span m="2695230">sort</span> <span m="2695560">of</span> <span m="2695620">imagine</span> <span m="2696100">that</span> <span m="2696250">might</span> <span m="2696450">be</span> <span m="2696550">true</span> <span m="2698000">is</span> <span m="2698570">that</span> <span m="2699720">I'm</span> <span m="2700090">getting</span> <span m="2700460">n</span> <span m="2700650">of</span> <span m="2700760">them,</span> <span m="2701040">so</span> <span m="2701080">I</span> <span m="2701120">might</span> <span m="2701320">be</span> <span m="2701410">multiplying--</span> <span m="2702120">the</span> <span m="2702270">top</span> <span m="2702580">one</span> <span m="2702770">is</span> <span m="2702910">n</span> <span m="2703060">squared,</span> <span m="2704500">so</span> <span m="2705270">I'm</span> <span m="2705450">going</span> <span m="2705590">to</span> <span m="2705650">have</span> <span m="2705840">n</span> <span m="2705906">of</span> <span m="2705973">them</span> <span m="2706240">about</span> <span m="2706600">n</span> <span m="2706750">squared.</span> <span m="2707900">Might</span> <span m="2708080">be</span> <span m="2708160">something</span> <span m="2708430">like</span> <span m="2708630">n</span> <span m="2708740">cubed.</span> <span m="2711370">That's</span> <span m="2711610">another</span> <span m="2711810">way</span> <span m="2712670">you</span> <span m="2712800">could</span> <span m="2712900">think</span> <span m="2713120">of</span> <span m="2713200">it,</span> <span m="2714230">to</span> <span m="2714340">guess</span> <span m="2714640">that.</span> <span m="2716660">Any</span> <span m="2716860">other</span> <span m="2717010">questions</span> <span m="2719560">on</span> <span m="2719700">this?</span> </p>
<p><span m="2725130">So</span> <span m="2725400">far,</span> <span m="2725620">all</span> <span m="2725960">these</span> <span m="2726190">sums</span> <span m="2726810">had</span> <span m="2727030">nice</span> <span m="2727310">closed</span> <span m="2727620">forms,</span> <span m="2729110">and</span> <span m="2729310">a</span> <span m="2729340">lot</span> <span m="2729600">of</span> <span m="2729670">them</span> <span m="2729970">do</span> <span m="2730350">that</span> <span m="2730480">you'll</span> <span m="2730640">encounter</span> <span m="2731190">later</span> <span m="2731520">on,</span> <span m="2732240">but</span> <span m="2732450">not</span> <span m="2732710">all,</span> <span m="2733410">and</span> <span m="2733640">sometimes</span> <span m="2734260">you</span> <span m="2734340">get</span> <span m="2734540">sums</span> <span m="2735510">that</span> <span m="2735750">don't</span> <span m="2736130">have</span> <span m="2736730">a</span> <span m="2736780">nice</span> <span m="2737020">closed</span> <span m="2737310">form--</span> <span m="2737580">at</span> <span m="2737650">least</span> <span m="2737880">nobody</span> <span m="2738210">has</span> <span m="2738320">ever</span> <span m="2738470">figured</span> <span m="2738790">out</span> <span m="2738980">one,</span> <span m="2739670">and</span> <span m="2740170">probably</span> <span m="2740720">doesn't</span> <span m="2741040">always</span> <span m="2741300">exist.</span> <span m="2742560">For</span> <span m="2742710">example,</span> <span m="2743280">what</span> <span m="2743570">if</span> <span m="2743710">I</span> <span m="2743940">want</span> <span m="2744200">to</span> <span m="2744260">sum</span> <span m="2745540">the</span> <span m="2745680">first</span> <span m="2746330">n</span> <span m="2746820">square</span> <span m="2747380">roots</span> <span m="2748390">of</span> <span m="2748550">integers?</span> <span m="2753340">Let's</span> <span m="2753490">write</span> <span m="2753918">that</span> <span m="2754346">down.</span> </p>
<p><span m="2764400">So</span> <span m="2764690">say</span> <span m="2764990">I</span> <span m="2765040">want</span> <span m="2765390">a</span> <span m="2765470">closed</span> <span m="2765990">form</span> <span m="2766520">for</span> <span m="2766920">this</span> <span m="2767120">guy.</span> <span m="2773560">Nobody</span> <span m="2773970">knows</span> <span m="2774870">an</span> <span m="2775030">answer</span> <span m="2775290">for</span> <span m="2775410">that,</span> <span m="2776940">but</span> <span m="2777150">there</span> <span m="2777340">are</span> <span m="2777640">ways</span> <span m="2778610">of</span> <span m="2778880">getting</span> <span m="2779240">very</span> <span m="2779750">good,</span> <span m="2780500">close</span> <span m="2780990">bounds</span> <span m="2781480">on</span> <span m="2781660">it</span> <span m="2781840">that</span> <span m="2782000">are</span> <span m="2782040">closed</span> <span m="2782410">form,</span> <span m="2782930">and</span> <span m="2783070">these</span> <span m="2783290">are</span> <span m="2783340">very</span> <span m="2783680">important,</span> <span m="2783950">and</span> <span m="2784220">we're</span> <span m="2784320">going</span> <span m="2784420">to</span> <span m="2784520">use</span> <span m="2784590">this</span> <span m="2785280">the</span> <span m="2785440">rest</span> <span m="2785660">of</span> <span m="2785730">today</span> <span m="2786060">and</span> <span m="2786150">the</span> <span m="2786220">rest</span> <span m="2786450">of</span> <span m="2786520">next</span> <span m="2786770">time.</span> <span m="2787890">And</span> <span m="2788280">they're</span> <span m="2788390">based</span> <span m="2788770">on</span> <span m="2789420">replacing</span> <span m="2790090">the</span> <span m="2790160">sum</span> <span m="2791040">with</span> <span m="2791230">an</span> <span m="2791320">integral,</span> <span m="2792740">and</span> <span m="2792890">the</span> <span m="2792990">integral</span> <span m="2793490">is</span> <span m="2793740">very</span> <span m="2794050">close</span> <span m="2794370">to</span> <span m="2794440">the</span> <span m="2794520">right</span> <span m="2794730">answer,</span> <span m="2795140">and</span> <span m="2795240">then</span> <span m="2795340">we</span> <span m="2795440">can</span> <span m="2795470">see</span> <span m="2795640">what</span> <span m="2795760">the</span> <span m="2795860">error</span> <span m="2796070">terms</span> <span m="2796370">are.</span> </p>
<p><span m="2797820">So</span> <span m="2798120">let's</span> <span m="2798340">first</span> <span m="2798680">look</span> <span m="2798810">at</span> <span m="2798880">the</span> <span m="2798950">case</span> <span m="2799390">when</span> <span m="2799580">we've</span> <span m="2799700">got</span> <span m="2799880">a</span> <span m="2799910">sum</span> <span m="2800270">where</span> <span m="2800370">the</span> <span m="2800460">terms</span> <span m="2800900">are</span> <span m="2801020">increasing</span> <span m="2801970">as</span> <span m="2802150">i</span> <span m="2802560">grows,</span> <span m="2806960">and</span> <span m="2807320">we'll</span> <span m="2807480">call</span> <span m="2807710">these</span> <span m="2808010">integration</span> <span m="2808650">bounds,</span> <span m="2814240">and</span> <span m="2814410">a</span> <span m="2814440">general</span> <span m="2814930">sum</span> <span m="2815145">will</span> <span m="2815360">look</span> <span m="2815590">like</span> <span m="2815800">this--</span> <span m="2816560">i</span> <span m="2816840">equals</span> <span m="2817000">1</span> <span m="2817230">to</span> <span m="2817380">n</span> <span m="2817790">of</span> <span m="2818140">f</span> <span m="2818400">of</span> <span m="2818550">i,</span> <span m="2820690">and</span> <span m="2820890">the</span> <span m="2820950">first</span> <span m="2821300">case</span> <span m="2822170">is</span> <span m="2822360">when</span> <span m="2822620">f</span> <span m="2822990">is</span> <span m="2823200">a</span> <span m="2823330">positive</span> <span m="2824330">increasing</span> <span m="2824870">function,</span> <span m="2832330">increasing</span> <span m="2832850">in</span> <span m="2832950">i.</span> <span m="2840660">Integration</span> <span m="2841040">bounds,</span> <span m="2841470">and</span> <span m="2841600">so</span> <span m="2841780">we're</span> <span m="2841990">increasing</span> <span m="2842475">function.</span> <span m="2848160">So</span> <span m="2848250">let</span> <span m="2848340">me</span> <span m="2848420">draw</span> <span m="2848670">a</span> <span m="2848720">picture</span> <span m="2849340">that</span> <span m="2849480">will</span> <span m="2849820">hopefully</span> <span m="2851530">make</span> <span m="2851790">the</span> <span m="2851860">bounds</span> <span m="2852145">that</span> <span m="2852430">we're</span> <span m="2852480">going</span> <span m="2852600">to</span> <span m="2852680">get</span> <span m="2853310">pretty</span> <span m="2853600">easy.</span> </p>
<p><span m="2867550">So</span> <span m="2868210">let's</span> <span m="2869080">draw</span> <span m="2870890">the</span> <span m="2871032">sum</span> <span m="2871460">here</span> <span m="2873130">as</span> <span m="2873360">follows.</span> <span m="2873590">I've</span> <span m="2873620">got</span> <span m="2873780">0,</span> <span m="2874320">1,</span> <span m="2875100">2,</span> <span m="2876320">3,</span> <span m="2879190">n</span> <span m="2879450">minus</span> <span m="2879810">2,</span> <span m="2880900">n</span> <span m="2881160">minus</span> <span m="2881510">1,</span> <span m="2882686">n,</span> <span m="2884360">and</span> <span m="2885430">draw</span> <span m="2885620">the</span> <span m="2885720">values</span> <span m="2886160">of</span> <span m="2886290">f</span> <span m="2886520">here.</span> <span m="2886840">Here's</span> <span m="2887110">f</span> <span m="2887410">of</span> <span m="2887640">1,</span> <span m="2889650">f</span> <span m="2889900">of</span> <span m="2890020">2--</span> <span m="2890620">it's</span> <span m="2890750">increasing--</span> <span m="2892900">f</span> <span m="2893030">of</span> <span m="2893160">3,</span> <span m="2897260">f</span> <span m="2897330">of</span> <span m="2897400">n</span> <span m="2897470">minus</span> <span m="2897820">1,</span> <span m="2898290">and</span> <span m="2898760">f</span> <span m="2899240">of</span> <span m="2899310">n.</span> <span m="2903180">Then</span> <span m="2903370">I'll</span> <span m="2903490">draw</span> <span m="2903960">the</span> <span m="2906430">rectangles</span> <span m="2907210">here.</span> <span m="2911640">So</span> <span m="2911970">this</span> <span m="2912200">has</span> <span m="2912520">area</span> <span m="2914290">of</span> <span m="2914675">f</span> <span m="2915060">of</span> <span m="2915160">1,</span> <span m="2916000">this</span> <span m="2916240">has</span> <span m="2916510">area</span> <span m="2916810">f</span> <span m="2917070">of</span> <span m="2917170">2,</span> <span m="2918620">this</span> <span m="2918850">has</span> <span m="2919080">area</span> <span m="2919370">f</span> <span m="2919590">of</span> <span m="2919670">3,</span> <span m="2921500">and</span> <span m="2921620">we</span> <span m="2921700">keep</span> <span m="2921950">on</span> <span m="2922160">going.</span> <span m="2923630">Let's</span> <span m="2924120">see,</span> <span m="2925620">this</span> <span m="2925840">will</span> <span m="2925930">be</span> <span m="2926100">f</span> <span m="2926240">of</span> <span m="2926360">n</span> <span m="2926480">minus</span> <span m="2926790">2</span> <span m="2927070">on</span> <span m="2927180">this</span> <span m="2927380">one--</span> <span m="2927750">I'll</span> <span m="2927868">just</span> <span m="2927986">do</span> <span m="2928460">f</span> <span m="2928530">of</span> <span m="2928605">n</span> <span m="2928680">minus</span> <span m="2928970">1,</span> <span m="2929210">draw</span> <span m="2929420">this</span> <span m="2929580">guy</span> <span m="2930420">here.</span> </p>
<p><span m="2933950">So</span> <span m="2934020">its</span> <span m="2934140">unit</span> <span m="2934510">width,</span> <span m="2934870">its</span> <span m="2934990">height</span> <span m="2935320">is</span> <span m="2935710">f</span> <span m="2935900">of</span> <span m="2935990">n</span> <span m="2936100">minus</span> <span m="2936400">1,</span> <span m="2936670">so</span> <span m="2936870">its</span> <span m="2937090">area</span> <span m="2937410">is</span> <span m="2937500">f</span> <span m="2937585">of</span> <span m="2937780">n</span> <span m="2937900">minus</span> <span m="2938200">1,</span> <span m="2939240">and</span> <span m="2939430">then</span> <span m="2942040">f</span> <span m="2942240">of</span> <span m="2942688">n.</span> <span m="2945690">And</span> <span m="2945780">let</span> <span m="2945880">me</span> <span m="2946010">also--</span> <span m="2946600">so</span> <span m="2946840">the</span> <span m="2947060">sum</span> <span m="2949460">of</span> <span m="2949890">f</span> <span m="2950110">of</span> <span m="2950280">i</span> <span m="2951240">is</span> <span m="2951500">the</span> <span m="2951720">areas</span> <span m="2953010">in</span> <span m="2953170">the</span> <span m="2953260">rectangles.</span> <span m="2954500">That's</span> <span m="2954730">what</span> <span m="2954830">the</span> <span m="2954890">sum</span> <span m="2955160">is,</span> <span m="2956300">and</span> <span m="2956440">I</span> <span m="2956500">want</span> <span m="2956760">to</span> <span m="2957300">get</span> <span m="2957480">bounds</span> <span m="2957930">on</span> <span m="2958070">this</span> <span m="2958330">sum</span> <span m="2959580">using</span> <span m="2959900">the</span> <span m="2960000">integral,</span> <span m="2961120">because</span> <span m="2961310">integrals</span> <span m="2961700">are</span> <span m="2961830">easier</span> <span m="2962120">to</span> <span m="2962190">compute.</span> </p>
<p><span m="2963510">So</span> <span m="2963690">let's</span> <span m="2964080">draw</span> <span m="2964450">the</span> <span m="2964540">function</span> <span m="2966670">f</span> <span m="2966890">of</span> <span m="2967020">x</span> <span m="2967890">from</span> <span m="2968070">1</span> <span m="2968310">to</span> <span m="2968410">n.</span> <span m="2975720">All</span> <span m="2975860">right,</span> <span m="2976030">so</span> <span m="2976160">this</span> <span m="2976470">is</span> <span m="2977920">f</span> <span m="2978140">of</span> <span m="2978280">x</span> <span m="2979120">as</span> <span m="2979290">a</span> <span m="2979330">function.</span> <span m="2982380">Now</span> <span m="2982630">I</span> <span m="2982740">claim</span> <span m="2984490">that</span> <span m="2985080">the</span> <span m="2985180">sum</span> <span m="2987480">i</span> <span m="2987790">equals</span> <span m="2988270">1</span> <span m="2988620">to</span> <span m="2988800">n</span> <span m="2989630">of</span> <span m="2989830">f</span> <span m="2990000">of</span> <span m="2990477">i</span> <span m="2991910">is</span> <span m="2992200">at</span> <span m="2992370">least</span> <span m="2993700">f</span> <span m="2993950">of</span> <span m="2994070">1</span> <span m="2994970">plus</span> <span m="2995380">the</span> <span m="2995560">integral</span> <span m="2996040">from</span> <span m="2996230">1</span> <span m="2996450">to</span> <span m="2996590">n,</span> <span m="2997610">f</span> <span m="2997840">of</span> <span m="2997970">x,</span> <span m="2998420">dx.</span> </p>
<p><span m="3001570">Now,</span> <span m="3001760">the</span> <span m="3001930">integral</span> <span m="3002960">from</span> <span m="3003240">1</span> <span m="3003670">to</span> <span m="3003790">n</span> <span m="3004810">of</span> <span m="3005030">f</span> <span m="3005260">of</span> <span m="3005400">x</span> <span m="3007590">is</span> <span m="3008030">this</span> <span m="3008220">stuff,</span> <span m="3008530">the</span> <span m="3008630">stuff</span> <span m="3008950">under</span> <span m="3009300">the</span> <span m="3009390">curve.</span> <span m="3010786">It</span> <span m="3011200">comes</span> <span m="3011490">down</span> <span m="3011790">here,</span> <span m="3013010">starts</span> <span m="3013590">at</span> <span m="3013680">1,</span> <span m="3014680">and</span> <span m="3014840">it's</span> <span m="3014930">the</span> <span m="3015010">stuff</span> <span m="3015270">under</span> <span m="3015430">the</span> <span m="3015530">curve.</span> <span m="3017760">And</span> <span m="3017790">what</span> <span m="3017900">I'm</span> <span m="3018010">saying</span> <span m="3018510">here</span> <span m="3018930">is</span> <span m="3019120">that</span> <span m="3019220">if</span> <span m="3019300">you</span> <span m="3019410">take</span> <span m="3019800">that</span> <span m="3020020">stuff</span> <span m="3020240">under</span> <span m="3020360">the</span> <span m="3020490">curve</span> <span m="3020930">and</span> <span m="3021080">add</span> <span m="3021490">f</span> <span m="3021690">of</span> <span m="3021800">1,</span> <span m="3022700">which</span> <span m="3022950">is</span> <span m="3023120">this</span> <span m="3023350">piece,</span> <span m="3024600">that's</span> <span m="3025000">a</span> <span m="3025180">lower</span> <span m="3025570">bound</span> <span m="3026710">on</span> <span m="3026890">our</span> <span m="3027030">sum.</span> <span m="3028230">The</span> <span m="3028290">sum's</span> <span m="3028620">bigger</span> <span m="3028870">than</span> <span m="3029000">that.</span> <span m="3031140">So</span> <span m="3031230">what</span> <span m="3031340">I'm</span> <span m="3031440">saying</span> <span m="3031840">is</span> <span m="3032020">the</span> <span m="3032180">area</span> <span m="3032620">in</span> <span m="3032690">the</span> <span m="3032790">rectangles</span> <span m="3034620">is</span> <span m="3034820">at</span> <span m="3034940">least</span> <span m="3035240">as</span> <span m="3035380">big</span> <span m="3035750">as</span> <span m="3036470">the</span> <span m="3036630">area</span> <span m="3036920">in</span> <span m="3036990">the</span> <span m="3037060">first</span> <span m="3037380">rectangle</span> <span m="3038440">plus</span> <span m="3039400">the</span> <span m="3039570">area</span> <span m="3040010">under</span> <span m="3040260">the</span> <span m="3040360">curve.</span> <span m="3042250">Does</span> <span m="3042280">everybody</span> <span m="3042520">see</span> <span m="3042650">why</span> <span m="3042800">that</span> <span m="3042990">is?</span> </p>
<p><span m="3045190">I'm</span> <span m="3045500">saying</span> <span m="3045810">the</span> <span m="3045870">sum</span> <span m="3046370">is</span> <span m="3046500">the</span> <span m="3046630">area</span> <span m="3046930">in</span> <span m="3047000">the</span> <span m="3047080">rectangles,</span> <span m="3049110">right?</span> <span m="3049340">That's</span> <span m="3049530">pretty</span> <span m="3049720">clear.</span> <span m="3051050">And</span> <span m="3051300">that</span> <span m="3051630">is</span> <span m="3052480">at</span> <span m="3052660">least</span> <span m="3053000">as</span> <span m="3053180">big</span> <span m="3053540">as</span> <span m="3054530">the</span> <span m="3054640">first</span> <span m="3055060">rectangle</span> <span m="3055570">f</span> <span m="3055670">of</span> <span m="3055790">1</span> <span m="3056760">plus</span> <span m="3057200">the</span> <span m="3057290">stuff</span> <span m="3057770">under</span> <span m="3058070">the</span> <span m="3058170">curve,</span> <span m="3059530">which</span> <span m="3059610">is</span> <span m="3059700">the</span> <span m="3059810">integral,</span> <span m="3060730">and</span> <span m="3060870">I've</span> <span m="3061030">left--</span> <span m="3061360">I've</span> <span m="3061560">chopped</span> <span m="3061970">off</span> <span m="3062310">these</span> <span m="3062550">guys.</span> <span m="3063040">That's</span> <span m="3063360">extra.</span> <span m="3064500">OK?</span> <span m="3066410">Is</span> <span m="3066530">that</span> <span m="3066680">all</span> <span m="3066780">right?</span> <span m="3067710">So</span> <span m="3067880">lower</span> <span m="3068110">bound.</span> <span m="3070200">Any</span> <span m="3070370">questions</span> <span m="3070820">on</span> <span m="3070920">the</span> <span m="3070990">lower</span> <span m="3071240">bound?</span> </p>
<p><span m="3072510">This</span> <span m="3072680">is</span> <span m="3072800">a</span> <span m="3072860">picture</span> <span m="3073230">proof,</span> <span m="3074520">which</span> <span m="3074750">we</span> <span m="3074910">always</span> <span m="3075120">tell</span> <span m="3075240">you</span> <span m="3075350">not</span> <span m="3075620">to</span> <span m="3075710">do,</span> <span m="3076120">but</span> <span m="3076460">we're</span> <span m="3076650">going</span> <span m="3076790">to</span> <span m="3076860">do</span> <span m="3077110">one</span> <span m="3078100">here.</span> <span m="3081830">And,</span> <span m="3081920">of</span> <span m="3081980">course,</span> <span m="3082940">it</span> <span m="3083240">totally</span> <span m="3083610">hides</span> <span m="3084810">why</span> <span m="3085180">did</span> <span m="3085310">I</span> <span m="3085370">need</span> <span m="3085650">f</span> <span m="3085830">is</span> <span m="3085970">increasing,</span> <span m="3086450">but</span> <span m="3086570">we'll</span> <span m="3086660">see</span> <span m="3086810">that</span> <span m="3086970">in</span> <span m="3087030">a</span> <span m="3087070">minute.</span> <span m="3087410">The</span> <span m="3087510">proof</span> <span m="3087770">would</span> <span m="3087870">not</span> <span m="3088090">work</span> <span m="3088360">unless</span> <span m="3088650">it</span> <span m="3088860">is</span> <span m="3089040">increasing</span> <span m="3089530">here.</span> <span m="3093310">Any</span> <span m="3093460">questions,</span> <span m="3093910">because</span> <span m="3094070">now</span> <span m="3094200">going</span> <span m="3094320">I'm</span> <span m="3094340">going</span> <span m="3094360">to</span> <span m="3094380">do</span> <span m="3094480">the</span> <span m="3094580">other</span> <span m="3094750">bound,</span> <span m="3095260">the</span> <span m="3095360">other</span> <span m="3095510">side.</span> </p>
<p><span m="3097390">I</span> <span m="3097680">also</span> <span m="3097960">claim--</span> <span m="3099170">this</span> <span m="3099320">will</span> <span m="3099380">be</span> <span m="3099440">a</span> <span m="3099490">little</span> <span m="3099670">trickier</span> <span m="3100000">to</span> <span m="3100080">see--</span> <span m="3101300">that</span> <span m="3101550">the</span> <span m="3101620">sum</span> <span m="3104190">is</span> <span m="3104400">at</span> <span m="3104530">most</span> <span m="3105720">f</span> <span m="3105950">of</span> <span m="3106040">n</span> <span m="3106830">plus</span> <span m="3107400">the</span> <span m="3107540">integral</span> <span m="3108460">from</span> <span m="3108630">1</span> <span m="3108810">to</span> <span m="3108890">n.</span> <span m="3113420">So</span> <span m="3113570">this</span> <span m="3113820">is</span> <span m="3113950">the</span> <span m="3114080">lower</span> <span m="3114420">bound</span> <span m="3115090">add</span> <span m="3115310">in</span> <span m="3115410">f</span> <span m="3115460">of</span> <span m="3115510">1,</span> <span m="3116110">the</span> <span m="3116240">upper</span> <span m="3116410">bound</span> <span m="3116770">just</span> <span m="3116970">add</span> <span m="3117100">in</span> <span m="3117180">f</span> <span m="3117260">of</span> <span m="3117340">n.</span> <span m="3117810">So</span> <span m="3117905">let's</span> <span m="3118000">see</span> <span m="3118130">why</span> <span m="3118320">that's</span> <span m="3118540">true.</span> <span m="3120440">Now,</span> <span m="3121610">to</span> <span m="3121740">see</span> <span m="3122040">that,</span> <span m="3122500">this</span> <span m="3122700">is--</span> <span m="3122890">I'm</span> <span m="3123020">not</span> <span m="3123140">going</span> <span m="3123270">to</span> <span m="3123470">be</span> <span m="3123572">able</span> <span m="3123675">to</span> <span m="3123777">draw</span> <span m="3123880">it.</span> <span m="3125400">I</span> <span m="3125510">want</span> <span m="3125760">you</span> <span m="3125840">to</span> <span m="3125940">imagine</span> <span m="3126480">taking</span> <span m="3126910">this</span> <span m="3127450">curve</span> <span m="3127910">and</span> <span m="3127980">the</span> <span m="3128090">area</span> <span m="3128420">under</span> <span m="3128740">it,</span> <span m="3129950">down</span> <span m="3130210">to</span> <span m="3130280">here,</span> <span m="3131990">and</span> <span m="3132170">sliding</span> <span m="3132790">it</span> <span m="3132930">left</span> <span m="3133360">one</span> <span m="3133640">unit.</span> </p>
<p><span m="3136480">sliding</span> <span m="3137170">it</span> <span m="3137300">left</span> <span m="3138950">to</span> <span m="3139100">here,</span> <span m="3140590">sliding</span> <span m="3141130">it</span> <span m="3141250">left</span> <span m="3141560">one</span> <span m="3141780">unit</span> <span m="3143180">over</span> <span m="3143390">to</span> <span m="3143470">here.</span> <span m="3146950">Now,</span> <span m="3147140">when</span> <span m="3147230">I</span> <span m="3147350">slide</span> <span m="3147850">it</span> <span m="3147970">left</span> <span m="3148820">one</span> <span m="3149120">unit,</span> <span m="3149480">did</span> <span m="3149840">the</span> <span m="3150150">area</span> <span m="3150570">under</span> <span m="3150890">it</span> <span m="3150980">change?</span> <span m="3153750">No.</span> <span m="3154280">It's</span> <span m="3154390">the</span> <span m="3154450">same</span> <span m="3154730">area</span> <span m="3155080">under</span> <span m="3155360">it,</span> <span m="3156360">just</span> <span m="3156650">where</span> <span m="3156860">it</span> <span m="3156950">sits</span> <span m="3157190">on</span> <span m="3157270">the</span> <span m="3157350">picture</span> <span m="3157760">is</span> <span m="3157910">now</span> <span m="3158920">out</span> <span m="3159355">here.</span> <span m="3162950">It's</span> <span m="3163140">this</span> <span m="3163410">area</span> <span m="3163760">under</span> <span m="3163980">this</span> <span m="3164190">guy,</span> <span m="3164400">but</span> <span m="3164530">it's</span> <span m="3164640">the</span> <span m="3164710">same</span> <span m="3165010">thing,</span> <span m="3165320">its</span> <span m="3165340">the</span> <span m="3165410">same</span> <span m="3165630">integral.</span> <span m="3167300">And</span> <span m="3167530">you</span> <span m="3167600">can</span> <span m="3167750">see</span> <span m="3168090">that</span> <span m="3168230">it's</span> <span m="3168690">more</span> <span m="3169230">than</span> <span m="3170390">what's</span> <span m="3170680">in</span> <span m="3170770">these</span> <span m="3170960">rectangles,</span> <span m="3171510">because</span> <span m="3171670">I</span> <span m="3171740">got</span> <span m="3171950">all</span> <span m="3172040">this</span> <span m="3172200">stuff.</span> </p>
<p><span m="3175110">And,</span> <span m="3175460">of</span> <span m="3175530">course,</span> <span m="3175940">I</span> <span m="3176020">didn't</span> <span m="3176200">even</span> <span m="3176360">include</span> <span m="3176770">this,</span> <span m="3177050">so</span> <span m="3177150">now</span> <span m="3177310">I</span> <span m="3177420">add</span> <span m="3177580">the</span> <span m="3177770">f</span> <span m="3178237">n.</span> <span m="3181040">So</span> <span m="3181190">if</span> <span m="3181270">I</span> <span m="3181350">take</span> <span m="3181610">the</span> <span m="3181770">area</span> <span m="3182320">under</span> <span m="3182580">the</span> <span m="3182680">curve,</span> <span m="3183230">which</span> <span m="3183280">is</span> <span m="3183370">the</span> <span m="3183510">integral,</span> <span m="3184640">shift</span> <span m="3184885">it</span> <span m="3185130">left</span> <span m="3185510">one,</span> <span m="3185890">so</span> <span m="3185965">it</span> <span m="3186040">only</span> <span m="3186190">goes</span> <span m="3186340">up</span> <span m="3186450">to</span> <span m="3186520">here</span> <span m="3186780">now,</span> <span m="3187430">and</span> <span m="3187600">then</span> <span m="3187770">add</span> <span m="3188090">in</span> <span m="3188260">this</span> <span m="3188440">rectangle,</span> <span m="3189950">that</span> <span m="3190240">dominates</span> <span m="3191100">the</span> <span m="3191240">area</span> <span m="3191820">in</span> <span m="3191940">the</span> <span m="3192020">rectangles.</span> <span m="3194590">Bigger</span> <span m="3194850">than.</span> <span m="3198890">Do</span> <span m="3199090">you</span> <span m="3199170">see</span> <span m="3199340">that?</span> <span m="3203400">I</span> <span m="3203520">could</span> <span m="3203660">do</span> <span m="3203760">a</span> <span m="3203830">lot</span> <span m="3204030">of</span> <span m="3204090">equations</span> <span m="3204600">on</span> <span m="3204660">the</span> <span m="3204730">board</span> <span m="3205020">but,</span> <span m="3205140">for</span> <span m="3205250">sure,</span> <span m="3205770">that</span> <span m="3205950">would</span> <span m="3206060">be</span> <span m="3206210">hopeless</span> <span m="3206720">to</span> <span m="3207170">follow.</span> <span m="3209550">Any</span> <span m="3209630">questions</span> <span m="3210170">about</span> <span m="3210430">this?</span> <span m="3211160">Yeah.</span> </p>
<p><span m="3211896">AUDIENCE:</span> <span m="3212352">I</span> <span m="3212808">guess</span> <span m="3213264">I</span> <span m="3213492">understand</span> <span m="3213530">the</span> <span m="3213606">lower</span> <span m="3213682">amounts</span> <span m="3213720">because</span> <span m="3214632">we're</span> <span m="3215088">cutting</span> <span m="3215544">off</span> <span m="3215696">the</span> <span m="3215848">triangles.</span> </p>
<p><span m="3216460">PROFESSOR: Yeah.</span> </p>
<p><span m="3218360">But</span> <span m="3220250">is</span> <span m="3220720">there--</span> <span m="3221190">is</span> <span m="3221660">it</span> <span m="3222160">a</span> <span m="3222285">lot</span> <span m="3222410">of</span> <span m="3222535">hand</span> <span m="3222660">waving,</span> <span m="3222760">or</span> <span m="3222860">am</span> <span m="3222960">I</span> <span m="3223060">just</span> <span m="3223160">missing</span> <span m="3223660">something,</span> <span m="3224160">that</span> <span m="3224326">it's</span> <span m="3224493">always</span> <span m="3224660">going</span> <span m="3224910">to</span> <span m="3225160">be</span> <span m="3225660">the</span> <span m="3226160">f</span> <span m="3226326">of</span> <span m="3226493">n</span> <span m="3226660">is</span> <span m="3227150">what</span> <span m="3227395">we--</span> </p>
<p><span m="3228130">PROFESSOR: Yeah.</span> <span m="3228670">There's</span> <span m="3228820">a</span> <span m="3228850">little</span> <span m="3229070">hand</span> <span m="3229290">waving</span> <span m="3229530">going</span> <span m="3229790">on,</span> <span m="3230750">but</span> <span m="3230870">I</span> <span m="3230940">do</span> <span m="3231140">believe</span> <span m="3231420">it</span> <span m="3231890">is</span> <span m="3231970">true.</span> <span m="3233090">With</span> <span m="3233340">equations</span> <span m="3233870">you</span> <span m="3233930">can</span> <span m="3234020">make</span> <span m="3234170">it</span> <span m="3234230">precise.</span> <span m="3234700">Let's</span> <span m="3234920">look</span> <span m="3235090">at</span> <span m="3235200">it</span> <span m="3235280">again,</span> <span m="3236320">do</span> <span m="3236450">this</span> <span m="3236650">one</span> <span m="3236810">more</span> <span m="3237000">time.</span> </p>
<p><span m="3237876">AUDIENCE:</span> <span m="3238314">[INAUDIBLE]</span> </p>
<p><span m="3239980">PROFESSOR: Yeah,</span> <span m="3240160">I'm</span> <span m="3240300">waving</span> <span m="3240550">hands</span> <span m="3240790">a</span> <span m="3240840">little</span> <span m="3241050">bit,</span> <span m="3241440">but</span> <span m="3241560">it</span> <span m="3241690">also--</span> <span m="3242280">hopefully</span> <span m="3242730">the</span> <span m="3242860">intuition</span> <span m="3243350">comes</span> <span m="3243580">across,</span> <span m="3244940">because</span> <span m="3247000">when</span> <span m="3247170">I</span> <span m="3247270">shift</span> <span m="3247720">left</span> <span m="3248610">by</span> <span m="3248780">one,</span> <span m="3250220">this</span> <span m="3250430">point</span> <span m="3250730">becomes</span> <span m="3251200">this</span> <span m="3251420">point,</span> <span m="3252360">and</span> <span m="3252510">that</span> <span m="3252700">point</span> <span m="3252920">becomes</span> <span m="3253230">that</span> <span m="3253450">point,</span> <span m="3254210">and</span> <span m="3254390">I'm</span> <span m="3254510">catching</span> <span m="3255890">sort</span> <span m="3256300">of</span> <span m="3256360">the</span> <span m="3256420">cusp</span> <span m="3256890">of</span> <span m="3257000">these</span> <span m="3257170">rectangles,</span> <span m="3258370">and</span> <span m="3258530">now</span> <span m="3258740">I</span> <span m="3258830">take</span> <span m="3259630">this</span> <span m="3259940">curve</span> <span m="3261160">here,</span> <span m="3264300">shifting</span> <span m="3264750">the</span> <span m="3264830">whole</span> <span m="3265080">curve</span> <span m="3265400">left</span> <span m="3265880">one</span> <span m="3266090">unit,</span> <span m="3268120">and</span> <span m="3268250">the</span> <span m="3268340">area</span> <span m="3268620">doesn't</span> <span m="3268860">change.</span> <span m="3270430">It</span> <span m="3270640">would</span> <span m="3270730">be</span> <span m="3270800">equivalent</span> <span m="3271300">of</span> <span m="3271440">taking</span> <span m="3271800">the</span> <span m="3271960">integral</span> <span m="3272510">from</span> <span m="3272700">0</span> <span m="3273010">to</span> <span m="3273090">n</span> <span m="3273290">minus</span> <span m="3273650">1.</span> <span m="3275170">You</span> <span m="3275300">notice</span> <span m="3275390">what</span> <span m="3275570">I'm</span> <span m="3275670">doing--</span> <span m="3275950">of</span> <span m="3276650">f</span> <span m="3276900">of</span> <span m="3277040">x</span> <span m="3277350">plus</span> <span m="3277710">1.</span> <span m="3279360">There's</span> <span m="3279450">another</span> <span m="3279660">way</span> <span m="3279760">to</span> <span m="3279830">look</span> <span m="3280020">at</span> <span m="3280110">it</span> <span m="3280200">mathematically,</span> <span m="3280850">maybe--</span> <span m="3281270">probably</span> <span m="3281640">the</span> <span m="3281700">formal</span> <span m="3282070">way</span> <span m="3282170">to</span> <span m="3282260">do</span> <span m="3282460">it--</span> <span m="3283410">but</span> <span m="3283630">the</span> <span m="3283720">idea</span> <span m="3284010">is</span> <span m="3284090">that</span> <span m="3284220">this</span> <span m="3284430">now</span> <span m="3284790">contains</span> <span m="3286060">all</span> <span m="3286460">these</span> <span m="3286740">first</span> <span m="3287070">n</span> <span m="3287210">minus</span> <span m="3287500">1</span> <span m="3287660">rectangles,</span> <span m="3288580">are</span> <span m="3288750">contained</span> <span m="3289250">underneath</span> <span m="3289680">it.</span> <span m="3291180">And</span> <span m="3291230">then</span> <span m="3291370">I</span> <span m="3291450">just</span> <span m="3291700">add</span> <span m="3291930">this</span> <span m="3292160">in,</span> <span m="3294170">and</span> <span m="3294880">I'm</span> <span m="3295060">good</span> <span m="3295190">to</span> <span m="3295270">go.</span> <span m="3295580">I've</span> <span m="3295730">contained</span> <span m="3296070">all</span> <span m="3296210">the</span> <span m="3296300">rectangles</span> <span m="3296930">now</span> <span m="3297370">for</span> <span m="3297516">the</span> <span m="3297663">upper</span> <span m="3297810">bounds.</span> <span m="3300690">All</span> <span m="3300750">right,</span> <span m="3300900">mathematically</span> <span m="3301500">what</span> <span m="3301650">this</span> <span m="3302035">is,</span> <span m="3302227">this</span> <span m="3302420">curve</span> <span m="3302800">is</span> <span m="3303770">0</span> <span m="3304180">to</span> <span m="3304270">n</span> <span m="3304440">minus</span> <span m="3304790">1,</span> <span m="3306270">fx</span> <span m="3306750">plus</span> <span m="3306990">1,</span> <span m="3308100">which</span> <span m="3308440">is</span> <span m="3308590">the</span> <span m="3308670">same</span> <span m="3308980">thing</span> <span m="3309260">as</span> <span m="3310000">what</span> <span m="3310170">that</span> <span m="3310350">is.</span> <span m="3313810">Any</span> <span m="3314360">questions</span> <span m="3316380">for</span> <span m="3316510">that?</span> </p>
<p><span m="3319180">But</span> <span m="3319280">now</span> <span m="3319460">I</span> <span m="3319560">have</span> <span m="3319910">good</span> <span m="3320080">bounds</span> <span m="3320660">on</span> <span m="3320930">the</span> <span m="3320990">sum.</span> <span m="3323160">We</span> <span m="3323260">know</span> <span m="3323490">that</span> <span m="3323670">the</span> <span m="3323740">sum</span> <span m="3324190">is</span> <span m="3324500">at</span> <span m="3324620">least</span> <span m="3325040">this</span> <span m="3325990">and</span> <span m="3326130">at</span> <span m="3326210">most</span> <span m="3326510">that,</span> <span m="3328100">and</span> <span m="3328300">those</span> <span m="3328620">two</span> <span m="3329830">bounds</span> <span m="3330320">only</span> <span m="3330540">differ</span> <span m="3330860">by</span> <span m="3331070">a</span> <span m="3331120">single</span> <span m="3331450">term</span> <span m="3333310">in</span> <span m="3333460">the</span> <span m="3333530">sum,</span> <span m="3333990">so</span> <span m="3334170">they're</span> <span m="3334310">very</span> <span m="3334610">close.</span> <span m="3336380">These</span> <span m="3336600">actually</span> <span m="3336910">are</span> <span m="3336950">good</span> <span m="3337120">formulas</span> <span m="3337540">to</span> <span m="3337630">write</span> <span m="3337830">down</span> <span m="3338040">for</span> <span m="3338130">your</span> <span m="3338220">crib</span> <span m="3338450">sheet.</span> <span m="3338810">That</span> <span m="3339050">is</span> <span m="3339200">worth</span> <span m="3339400">doing</span> <span m="3341850">on</span> <span m="3342040">the</span> <span m="3342110">test,</span> <span m="3345100">and</span> <span m="3345280">we'll</span> <span m="3345570">use</span> <span m="3345830">them</span> <span m="3347270">more</span> <span m="3348350">today,</span> <span m="3348880">and</span> <span m="3349080">we'll</span> <span m="3349170">also</span> <span m="3349450">use</span> <span m="3349650">them</span> <span m="3350670">on</span> <span m="3350870">Thursday.</span> </p>
<p><span m="3356730">So</span> <span m="3356870">now</span> <span m="3357170">we</span> <span m="3357280">can</span> <span m="3357450">actually</span> <span m="3358030">get</span> <span m="3358280">close</span> <span m="3358770">bounds</span> <span m="3359400">on</span> <span m="3359670">the</span> <span m="3359740">sum</span> <span m="3360030">of</span> <span m="3360090">the</span> <span m="3360150">square</span> <span m="3360500">roots.</span> <span m="3362850">So</span> <span m="3362960">let's</span> <span m="3363140">see</span> <span m="3363290">how</span> <span m="3363420">this</span> <span m="3363620">works</span> <span m="3364030">for</span> <span m="3366200">f</span> <span m="3366365">of</span> <span m="3366530">i</span> <span m="3367120">equal</span> <span m="3367480">to</span> <span m="3367520">the</span> <span m="3367560">square</span> <span m="3367820">root</span> <span m="3367920">of</span> <span m="3368060">i,</span> <span m="3368890">which</span> <span m="3369160">is</span> <span m="3369340">increasing.</span> <span m="3371120">So</span> <span m="3371300">I</span> <span m="3371350">compute</span> <span m="3371720">the</span> <span m="3371850">integral</span> <span m="3372210">from</span> <span m="3372400">1</span> <span m="3372630">to</span> <span m="3373150">n</span> <span m="3373620">of</span> <span m="3373900">square</span> <span m="3374350">root</span> <span m="3374580">of</span> <span m="3374830">x</span> <span m="3375307">dx,</span> <span m="3377470">and</span> <span m="3377600">that's</span> <span m="3377800">just</span> <span m="3378190">x</span> <span m="3378430">to</span> <span m="3378510">the</span> <span m="3378600">3/2</span> <span m="3379620">over</span> <span m="3380580">3/2,</span> <span m="3381830">evaluated</span> <span m="3382590">at</span> <span m="3382750">n</span> <span m="3382920">and</span> <span m="3383090">1,</span> <span m="3384960">and</span> <span m="3385110">that</span> <span m="3385340">just</span> <span m="3385445">equals</span> <span m="3385550">2/3</span> <span m="3388260">n</span> <span m="3388580">to</span> <span m="3388650">the</span> <span m="3388720">3/2</span> <span m="3389820">minus</span> <span m="3390185">1.</span> </p>
<p><span m="3392660">And</span> <span m="3392830">now</span> <span m="3393060">I</span> <span m="3393140">can</span> <span m="3393290">compute</span> <span m="3393620">the</span> <span m="3393690">bounds</span> <span m="3394250">on</span> <span m="3395080">the</span> <span m="3395200">sum</span> <span m="3395820">of</span> <span m="3395950">the</span> <span m="3396010">first</span> <span m="3396370">n</span> <span m="3396530">square</span> <span m="3396820">roots.</span> <span m="3399920">So</span> <span m="3400070">I</span> <span m="3400150">know</span> <span m="3400490">that</span> <span m="3401860">square</span> <span m="3402290">root</span> <span m="3403040">i</span> <span m="3403390">equals</span> <span m="3403740">1</span> <span m="3404000">to</span> <span m="3404180">n--</span> <span m="3407600">well,</span> <span m="3407750">the</span> <span m="3407890">upper</span> <span m="3408100">bound</span> <span m="3408560">is</span> <span m="3409000">f</span> <span m="3409473">n</span> <span m="3410420">plus</span> <span m="3411060">the</span> <span m="3411210">integral,</span> <span m="3417000">and</span> <span m="3417150">the</span> <span m="3417210">lower</span> <span m="3417560">bound</span> <span m="3418250">is</span> <span m="3418490">f</span> <span m="3418710">of</span> <span m="3418810">1</span> <span m="3420540">plus</span> <span m="3421620">the</span> <span m="3421770">integral.</span> <span m="3427590">What</span> <span m="3427850">is</span> <span m="3428710">f</span> <span m="3428930">of</span> <span m="3429020">n?</span> <span m="3432280">That's</span> <span m="3432510">the</span> <span m="3432580">square</span> <span m="3432850">root</span> <span m="3432970">of</span> <span m="3433080">n.</span> <span m="3435150">What</span> <span m="3435390">is</span> <span m="3435520">f</span> <span m="3435710">of</span> <span m="3435820">1?</span> <span m="3438330">The</span> <span m="3438760">square</span> <span m="3439030">root</span> <span m="3439130">of</span> <span m="3439200">1,</span> <span m="3439510">which</span> <span m="3439690">is</span> <span m="3439800">1.</span> <span m="3441860">So</span> <span m="3442300">I'll</span> <span m="3442400">plug</span> <span m="3443590">that</span> <span m="3443880">in</span> <span m="3444060">here.</span> <span m="3444310">I</span> <span m="3444390">get</span> <span m="3445200">2/3</span> <span m="3446890">n</span> <span m="3447140">to</span> <span m="3447210">the</span> <span m="3447290">3/2</span> <span m="3449040">plus</span> <span m="3449410">1</span> <span m="3449660">minus</span> <span m="3450080">2/3</span> <span m="3450770">is</span> <span m="3450880">plus</span> <span m="3451170">1/3,</span> <span m="3454440">and</span> <span m="3454690">here</span> <span m="3455060">I</span> <span m="3455150">get</span> <span m="3456050">2/3</span> <span m="3457680">n</span> <span m="3457940">to</span> <span m="3458000">the</span> <span m="3458060">3/2</span> <span m="3460130">plus</span> <span m="3460560">the</span> <span m="3460640">square</span> <span m="3460900">root</span> <span m="3461020">of</span> <span m="3461150">n</span> <span m="3461830">minus</span> <span m="3462360">2/3.</span> <span m="3466210">So</span> <span m="3466920">now</span> <span m="3467280">I</span> <span m="3467400">have</span> <span m="3467630">pretty</span> <span m="3467890">good</span> <span m="3468110">bounds</span> <span m="3468710">on</span> <span m="3468830">the</span> <span m="3468900">sum</span> <span m="3469170">of</span> <span m="3469240">the</span> <span m="3469300">first</span> <span m="3469670">n</span> <span m="3469840">square</span> <span m="3470080">roots</span> <span m="3470870">using</span> <span m="3471190">this</span> <span m="3471370">method.</span> </p>
<p><span m="3474320">So</span> <span m="3474540">for</span> <span m="3474680">example,</span> <span m="3476070">take</span> <span m="3476660">n</span> <span m="3476830">equals</span> <span m="3477070">100.</span> <span m="3483180">This</span> <span m="3483520">number</span> <span m="3483940">evaluates</span> <span m="3484540">to</span> <span m="3484650">667.</span> <span m="3487270">This</span> <span m="3487530">number</span> <span m="3487830">evaluates</span> <span m="3488420">to</span> <span m="3488540">676,</span> <span m="3491110">and</span> <span m="3491380">the</span> <span m="3491440">difference</span> <span m="3492130">is</span> <span m="3492430">9,</span> <span m="3492950">which</span> <span m="3493210">is</span> <span m="3493360">the</span> <span m="3493440">square</span> <span m="3493720">root</span> <span m="3493820">of</span> <span m="3493880">100</span> <span m="3494180">minus</span> <span m="3494480">1.</span> <span m="3496290">This</span> <span m="3496570">is</span> <span m="3497780">the</span> <span m="3497840">square</span> <span m="3498060">root</span> <span m="3498160">of</span> <span m="3498250">n,</span> <span m="3499210">this</span> <span m="3499450">is</span> <span m="3499570">1,</span> <span m="3500010">so</span> <span m="3500140">the</span> <span m="3500240">gap</span> <span m="3500590">here,</span> <span m="3500880">square</span> <span m="3501050">root</span> <span m="3501220">of</span> <span m="3501320">n</span> <span m="3501460">minus</span> <span m="3501800">1,</span> <span m="3503150">and</span> <span m="3503350">n</span> <span m="3503480">equals</span> <span m="3503700">100.</span> <span m="3504210">Square</span> <span m="3504340">root</span> <span m="3504470">of</span> <span m="3504530">100</span> <span m="3504800">minus</span> <span m="3505080">1</span> <span m="3505230">is</span> <span m="3505360">9,</span> <span m="3505830">so</span> <span m="3506020">I</span> <span m="3506110">shouldn't</span> <span m="3506370">be</span> <span m="3506430">surprised</span> <span m="3506840">my</span> <span m="3506950">gap</span> <span m="3507340">here</span> <span m="3507470">is</span> <span m="3507590">nine.</span> </p>
<p><span m="3508670">So</span> <span m="3508820">I</span> <span m="3508880">didn't</span> <span m="3509060">get</span> <span m="3509170">exactly</span> <span m="3509650">the</span> <span m="3509760">right</span> <span m="3509970">answer,</span> <span m="3510350">but</span> <span m="3510510">I'm</span> <span m="3510910">pretty</span> <span m="3511140">close</span> <span m="3512560">here.</span> <span m="3516030">Now,</span> <span m="3517420">as</span> <span m="3517750">n</span> <span m="3518030">grows,</span> <span m="3521200">what</span> <span m="3521400">happens</span> <span m="3521770">to</span> <span m="3521860">the</span> <span m="3521960">gap</span> <span m="3522370">between</span> <span m="3523210">the</span> <span m="3523480">upper</span> <span m="3523650">bound</span> <span m="3524670">and</span> <span m="3524830">the</span> <span m="3524900">lower</span> <span m="3525160">bound?</span> <span m="3527820">What</span> <span m="3527940">does</span> <span m="3528040">it</span> <span m="3528120">do?</span> <span m="3529370">It</span> <span m="3529500">gets</span> <span m="3529700">bigger.</span> <span m="3531000">So</span> <span m="3531460">my</span> <span m="3531610">gap</span> <span m="3531960">gets</span> <span m="3532170">bigger.</span> <span m="3533010">That's</span> <span m="3533220">not</span> <span m="3533410">so</span> <span m="3533600">nice.</span> <span m="3534490">Doesn't</span> <span m="3534770">always</span> <span m="3535000">stay</span> <span m="3535245">9</span> <span m="3535490">forever,</span> <span m="3537320">but</span> <span m="3538980">somehow</span> <span m="3539610">though</span> <span m="3540440">this</span> <span m="3540660">is</span> <span m="3540750">still</span> <span m="3541000">pretty</span> <span m="3541240">good,</span> <span m="3541590">because</span> <span m="3543890">the</span> <span m="3544020">gap</span> <span m="3544350">grows</span> <span m="3544890">but</span> <span m="3544990">the</span> <span m="3545080">gap</span> <span m="3545360">only</span> <span m="3545560">grows</span> <span m="3546180">as</span> <span m="3547130">square</span> <span m="3547440">root</span> <span m="3547560">of</span> <span m="3547650">n,</span> <span m="3549420">where</span> <span m="3550150">the</span> <span m="3550370">answer--</span> <span m="3550800">the</span> <span m="3550870">bounds--</span> <span m="3551135">are</span> <span m="3551400">growing</span> <span m="3551850">as</span> <span m="3552160">n</span> <span m="3552420">to</span> <span m="3552480">the</span> <span m="3552550">3/2.</span> </p>
<p><span m="3554670">In</span> <span m="3554820">other</span> <span m="3554990">words,</span> <span m="3555390">my</span> <span m="3555560">error</span> <span m="3557400">is</span> <span m="3557590">somewhere</span> <span m="3558000">around</span> <span m="3558350">here,</span> <span m="3559500">and</span> <span m="3559570">that</span> <span m="3559760">gets</span> <span m="3559980">smaller</span> <span m="3561090">compared</span> <span m="3561660">to</span> <span m="3561790">my</span> <span m="3562020">answer,</span> <span m="3563060">which</span> <span m="3563150">is</span> <span m="3563240">somewhere</span> <span m="3563610">around</span> <span m="3563900">there.</span> <span m="3565970">And</span> <span m="3566060">there's</span> <span m="3566210">a</span> <span m="3566270">special</span> <span m="3566770">notation</span> <span m="3567450">that</span> <span m="3567560">people</span> <span m="3567860">use.</span> <span m="3570010">In</span> <span m="3570460">fact,</span> <span m="3571810">let's</span> <span m="3572010">write</span> <span m="3572240">that</span> <span m="3572400">down</span> <span m="3572610">and</span> <span m="3572700">then</span> <span m="3572830">do</span> <span m="3572930">the</span> <span m="3573010">notation.</span> </p>
<p><span m="3585570">Another</span> <span m="3585830">way</span> <span m="3585990">of</span> <span m="3586110">writing</span> <span m="3586510">this</span> <span m="3586760">is</span> <span m="3588180">that</span> <span m="3588360">the</span> <span m="3590910">sum</span> <span m="3592835">i</span> <span m="3593240">equals</span> <span m="3593610">1</span> <span m="3593950">to</span> <span m="3594110">n</span> <span m="3594440">of</span> <span m="3594500">square</span> <span m="3594740">root</span> <span m="3594850">of</span> <span m="3595030">i.</span> <span m="3596820">The</span> <span m="3597000">leading</span> <span m="3597400">term</span> <span m="3597850">here</span> <span m="3598210">is</span> <span m="3598390">2/3</span> <span m="3598940">n</span> <span m="3599080">to</span> <span m="3599170">the</span> <span m="3599230">3/2,</span> <span m="3602670">and</span> <span m="3602840">then</span> <span m="3603000">there's</span> <span m="3603230">some</span> <span m="3603660">error</span> <span m="3604010">term--</span> <span m="3604450">delta</span> <span m="3604770">term</span> <span m="3605120">here.</span> <span m="3605630">We'll</span> <span m="3606060">call</span> <span m="3606360">that</span> <span m="3606990">delta</span> <span m="3607370">n,</span> <span m="3609020">and</span> <span m="3609330">we</span> <span m="3609430">know</span> <span m="3609830">that</span> <span m="3613470">the</span> <span m="3613660">error</span> <span m="3613970">term</span> <span m="3614430">is</span> <span m="3614540">at</span> <span m="3614640">least</span> <span m="3614950">1/3</span> <span m="3617640">and,</span> <span m="3617880">at</span> <span m="3617980">most,</span> <span m="3619190">the</span> <span m="3619220">square</span> <span m="3619340">root</span> <span m="3619460">of</span> <span m="3619600">n</span> <span m="3619740">minus</span> <span m="3620120">2/3.</span> <span m="3625100">So</span> <span m="3625330">this</span> <span m="3625620">delta</span> <span m="3626040">term</span> <span m="3627220">is</span> <span m="3627430">bound</span> <span m="3627810">by</span> <span m="3627950">the</span> <span m="3628000">square</span> <span m="3628280">root</span> <span m="3628350">of</span> <span m="3628390">n.</span> <span m="3629090">That's</span> <span m="3629330">getting</span> <span m="3629610">bigger</span> <span m="3629920">as</span> <span m="3630040">n</span> <span m="3630160">gets</span> <span m="3630370">big,</span> <span m="3631220">but</span> <span m="3632480">this</span> <span m="3632770">value</span> <span m="3633210">compared</span> <span m="3633720">to</span> <span m="3633780">your</span> <span m="3633940">answer</span> <span m="3634790">is</span> <span m="3634930">getting</span> <span m="3635160">small.</span> <span m="3636920">That's</span> <span m="3637260">nice,</span> <span m="3637990">and</span> <span m="3638160">so</span> <span m="3638290">the</span> <span m="3638380">way</span> <span m="3638740">that</span> <span m="3638920">gets</span> <span m="3639130">represented</span> <span m="3640530">is</span> <span m="3641330">as</span> <span m="3641510">follows.</span> </p>
<p><span m="3642050">We</span> <span m="3642060">would</span> <span m="3642200">say--</span> <span m="3642690">it's</span> <span m="3642780">using</span> <span m="3643040">that</span> <span m="3643200">tilde</span> <span m="3643550">notation.</span> <span m="3646620">We</span> <span m="3646770">write</span> <span m="3647020">tilde</span> <span m="3649000">2/3</span> <span m="3650000">times</span> <span m="3650230">n</span> <span m="3650500">to</span> <span m="3650560">the</span> <span m="3650640">3/2,</span> <span m="3651960">and</span> <span m="3652110">now</span> <span m="3652280">we've</span> <span m="3652450">gotten</span> <span m="3652790">rid</span> <span m="3652990">of</span> <span m="3653060">the</span> <span m="3653140">delta,</span> <span m="3653620">because</span> <span m="3654020">this</span> <span m="3654200">tilde</span> <span m="3654620">is</span> <span m="3654730">telling</span> <span m="3655110">us</span> <span m="3655260">that</span> <span m="3655420">everything</span> <span m="3655830">else</span> <span m="3656060">out</span> <span m="3656200">here</span> <span m="3657550">gets</span> <span m="3657810">small</span> <span m="3658510">compared</span> <span m="3658950">to</span> <span m="3659020">this</span> <span m="3659630">as</span> <span m="3659840">n</span> <span m="3660060">gets</span> <span m="3660270">big.</span> <span m="3661500">And</span> <span m="3661660">the</span> <span m="3661730">formal</span> <span m="3662180">definition--</span> <span m="3663800">Let's</span> <span m="3663910">write</span> <span m="3664140">out</span> <span m="3664260">the</span> <span m="3664330">formal</span> <span m="3664720">definition</span> <span m="3665157">for</span> <span m="3665594">it.</span> <span m="3682720">Now,</span> <span m="3682890">a</span> <span m="3682930">lot</span> <span m="3683120">of</span> <span m="3683190">times</span> <span m="3683630">you'll</span> <span m="3683780">see</span> <span m="3684030">people</span> <span m="3684670">use</span> <span m="3685110">this</span> <span m="3685290">symbol</span> <span m="3685600">to</span> <span m="3685670">mean</span> <span m="3685990">about.</span> <span m="3687170">That's</span> <span m="3687440">informal.</span> <span m="3688790">When</span> <span m="3688990">I'm</span> <span m="3689130">using</span> <span m="3689430">it</span> <span m="3689520">here,</span> <span m="3690330">it's</span> <span m="3690600">a</span> <span m="3690640">very</span> <span m="3690970">formal</span> <span m="3691440">meaning</span> <span m="3692180">mathematically.</span> <span m="3695580">A</span> <span m="3695670">function</span> <span m="3696190">g</span> <span m="3696450">of</span> <span m="3696570">x</span> <span m="3697340">is</span> <span m="3697480">tilde,</span> <span m="3698140">a</span> <span m="3698240">function</span> <span m="3698660">h</span> <span m="3698990">of</span> <span m="3699100">x</span> <span m="3702930">means</span> <span m="3703780">that</span> <span m="3703930">the</span> <span m="3704010">limit</span> <span m="3705810">as</span> <span m="3706250">x</span> <span m="3706620">goes</span> <span m="3707060">to</span> <span m="3707380">infinity</span> <span m="3708700">of</span> <span m="3709090">g</span> <span m="3710860">over</span> <span m="3711260">h</span> <span m="3713700">is</span> <span m="3713890">1.</span> </p>
<p><span m="3716840">In</span> <span m="3716990">other</span> <span m="3717160">words,</span> <span m="3717560">that</span> <span m="3717660">as</span> <span m="3717740">x</span> <span m="3718030">goes</span> <span m="3718290">to</span> <span m="3718380">infinity--</span> <span m="3718750">as</span> <span m="3718880">x</span> <span m="3719010">gets</span> <span m="3719260">big--</span> <span m="3719900">the</span> <span m="3720050">ratio</span> <span m="3720570">of</span> <span m="3720650">these</span> <span m="3720870">guys</span> <span m="3721190">becomes</span> <span m="3721550">1.</span> <span m="3724480">And</span> <span m="3724570">let's</span> <span m="3724750">see</span> <span m="3724900">if</span> <span m="3725000">that's</span> <span m="3725220">true</span> <span m="3725520">here.</span> <span m="3729040">Well,</span> <span m="3731270">square</span> <span m="3731680">root</span> <span m="3731720">of</span> <span m="3731760">i</span> <span m="3731900">equals</span> <span m="3732540">this,</span> <span m="3732930">so</span> <span m="3733090">I</span> <span m="3733170">need</span> <span m="3733330">to</span> <span m="3733400">show</span> <span m="3733610">the</span> <span m="3733700">limit</span> <span m="3734570">of</span> <span m="3734760">this</span> <span m="3735750">over</span> <span m="3736060">that</span> <span m="3736730">is</span> <span m="3736930">1.</span> <span m="3737730">So</span> <span m="3737910">let's</span> <span m="3738130">check</span> <span m="3738390">that.</span> <span m="3740550">A</span> <span m="3740610">limit</span> <span m="3740920">as</span> <span m="3741070">n</span> <span m="3741220">goes</span> <span m="3741440">to</span> <span m="3741540">infinity</span> <span m="3743540">of</span> <span m="3743840">2/3</span> <span m="3745180">n</span> <span m="3745440">to</span> <span m="3745500">the</span> <span m="3745570">3/2</span> <span m="3746360">plus</span> <span m="3746950">that</span> <span m="3747920">delta</span> <span m="3748250">term</span> <span m="3749690">over</span> <span m="3750810">2/3</span> <span m="3751520">n</span> <span m="3751670">to</span> <span m="3751740">the</span> <span m="3751800">3/2.</span> <span m="3753710">Well,</span> <span m="3754020">that</span> <span m="3754350">equals--</span> <span m="3755950">divide</span> <span m="3756360">out</span> <span m="3756710">by</span> <span m="3756940">2/3</span> <span m="3757390">n</span> <span m="3757530">to</span> <span m="3757565">the</span> <span m="3757600">3/2,</span> <span m="3758050">I</span> <span m="3758120">get</span> <span m="3758250">a</span> <span m="3758310">1.</span> </p>
<p><span m="3766100">Did</span> <span m="3766230">I--</span> <span m="3766330">should</span> <span m="3766580">I</span> <span m="3766690">have</span> <span m="3766800">subtracted?</span> <span m="3767150">No,</span> <span m="3767500">that's</span> <span m="3767966">OK.</span> <span m="3769830">One</span> <span m="3770500">plus</span> <span m="3772160">delta</span> <span m="3772620">n</span> <span m="3773160">over</span> <span m="3774060">2/3</span> <span m="3774870">n</span> <span m="3775030">to</span> <span m="3775090">the</span> <span m="3775160">3/2.</span> <span m="3777850">If</span> <span m="3777930">I</span> <span m="3777980">can</span> <span m="3778110">pull</span> <span m="3778320">the</span> <span m="3778390">1</span> <span m="3778620">out</span> <span m="3778780">front,</span> <span m="3779100">that's</span> <span m="3779320">1</span> <span m="3779680">plus</span> <span m="3780190">the</span> <span m="3780300">limit,</span> <span m="3782090">and</span> <span m="3782320">this</span> <span m="3782560">is</span> <span m="3782720">now</span> <span m="3783270">delta</span> <span m="3783610">n</span> <span m="3783830">is</span> <span m="3783930">at</span> <span m="3784000">most</span> <span m="3784230">square</span> <span m="3784490">root</span> <span m="3784600">of</span> <span m="3784740">n,</span> <span m="3785840">so</span> <span m="3785990">I</span> <span m="3786050">get</span> <span m="3786230">square</span> <span m="3786570">root</span> <span m="3786615">of</span> <span m="3786760">n</span> <span m="3787580">over</span> <span m="3788550">2/3</span> <span m="3789350">n</span> <span m="3789550">to</span> <span m="3789610">the</span> <span m="3789670">3/2.</span> <span m="3791860">Square</span> <span m="3792250">root</span> <span m="3792350">of</span> <span m="3792440">n</span> <span m="3792720">over</span> <span m="3792980">n</span> <span m="3793110">to</span> <span m="3793180">the</span> <span m="3793250">3/2,</span> <span m="3794050">that</span> <span m="3794350">goes</span> <span m="3794670">to</span> <span m="3794790">0.</span> <span m="3797110">So</span> <span m="3797350">this</span> <span m="3797620">equals</span> <span m="3798050">1.</span> <span m="3802280">So</span> <span m="3802390">this</span> <span m="3802640">limit</span> <span m="3802950">is</span> <span m="3803150">1,</span> <span m="3803670">and</span> <span m="3803800">so</span> <span m="3803920">therefore</span> <span m="3804370">I</span> <span m="3804440">can</span> <span m="3804600">say</span> <span m="3805030">that</span> <span m="3806130">the</span> <span m="3806240">sum</span> <span m="3806500">of</span> <span m="3806580">the</span> <span m="3806640">first</span> <span m="3807170">n</span> <span m="3807420">square</span> <span m="3807720">roots</span> <span m="3808140">is</span> <span m="3808360">tilde</span> <span m="3809570">2/3</span> <span m="3810090">n</span> <span m="3810230">to</span> <span m="3810290">the</span> <span m="3810350">3/2.</span> <span m="3812500">Any</span> <span m="3812670">questions</span> <span m="3814890">about</span> <span m="3815150">this?</span> <span m="3815320">We're</span> <span m="3815380">going</span> <span m="3815510">to</span> <span m="3815570">do</span> <span m="3815630">a</span> <span m="3815690">lot</span> <span m="3816100">of</span> <span m="3816160">this</span> <span m="3816300">kind</span> <span m="3816450">of</span> <span m="3816510">notation</span> <span m="3817400">next</span> <span m="3817570">time.</span> <span m="3818050">Yeah.</span> </p>
<p><span m="3819274">AUDIENCE:</span> <span m="3819756">When you took</span> <span m="3820240">the integral</span> <span m="3820690">and</span> <span m="3821140">got</span> <span m="3821530">2/3</span> <span m="3821670">into</span> <span m="3821930">the</span> <span m="3822400">3/2</span> <span m="3822760">minus</span> <span m="3823000">1,</span> <span m="3824460">why</span> <span m="3824850">did</span> <span m="3825080">the</span> <span m="3825450">minus</span> <span m="3825840">1</span> <span m="3827170">not</span> <span m="3827400">become</span> <span m="3827720">part</span> <span m="3827910">of</span> <span m="3828000">the</span> <span m="3828620">actual</span> <span m="3828880">solution</span> <span m="3829050">and</span> <span m="3829220">become</span> <span m="3829530">part</span> <span m="3829615">of</span> <span m="3829657">the</span> <span m="3829700">delta?</span> </p>
<p><span m="3831430">PROFESSOR: OK.</span> <span m="3831900">So</span> <span m="3832330">you</span> <span m="3832530">brought</span> <span m="3832770">it</span> <span m="3832875">up</span> <span m="3832980">to</span> <span m="3833060">here,</span> <span m="3833330">right?</span> <span m="3834500">OK.</span> <span m="3835360">So</span> <span m="3836250">then</span> <span m="3836530">I</span> <span m="3836620">plug</span> <span m="3837660">that</span> <span m="3838220">into</span> <span m="3838880">the</span> <span m="3839030">integral</span> <span m="3839220">that</span> <span m="3839410">appears</span> <span m="3839840">on</span> <span m="3839930">both</span> <span m="3840150">sides</span> <span m="3840490">here,</span> <span m="3840640">and</span> <span m="3840730">here</span> <span m="3840920">I</span> <span m="3841060">add</span> <span m="3841190">the</span> <span m="3841290">f</span> <span m="3841480">1,</span> <span m="3841790">here</span> <span m="3841970">I</span> <span m="3842070">add</span> <span m="3842300">f</span> <span m="3842605">n,</span> <span m="3844200">and</span> <span m="3844380">now</span> <span m="3844580">I</span> <span m="3844680">have</span> <span m="3845450">the</span> <span m="3845610">lower</span> <span m="3845890">bound</span> <span m="3846240">here</span> <span m="3846720">and</span> <span m="3846830">the</span> <span m="3846910">upper</span> <span m="3847060">bound</span> <span m="3847320">here.</span> <span m="3847480">Are</span> <span m="3847570">you</span> <span m="3847650">good</span> <span m="3847820">with</span> <span m="3847930">those?</span> <span m="3849150">All</span> <span m="3849370">right.</span> </p>
<p><span m="3849590">Now</span> <span m="3849980">some</span> <span m="3850170">judgment</span> <span m="3850620">takes</span> <span m="3850890">place,</span> <span m="3852100">and</span> <span m="3852340">what</span> <span m="3852450">I'm</span> <span m="3852590">really</span> <span m="3852880">trying</span> <span m="3853150">to</span> <span m="3853230">do</span> <span m="3853490">here</span> <span m="3854310">is</span> <span m="3854540">figure</span> <span m="3854830">out</span> <span m="3855020">what</span> <span m="3855310">are</span> <span m="3855400">the</span> <span m="3855520">important</span> <span m="3856100">terms</span> <span m="3857590">in</span> <span m="3857790">these</span> <span m="3857980">bounds</span> <span m="3858820">as</span> <span m="3859050">n</span> <span m="3859200">gets</span> <span m="3859410">big?</span> <span m="3860250">How</span> <span m="3860530">big</span> <span m="3860800">is</span> <span m="3860970">this</span> <span m="3861260">growing</span> <span m="3861730">as</span> <span m="3861870">n</span> <span m="3862070">gets</span> <span m="3862250">big?</span> <span m="3863440">Well,</span> <span m="3863570">as</span> <span m="3863700">n</span> <span m="3863890">gets</span> <span m="3864060">big,</span> <span m="3865240">the</span> <span m="3865340">1/3</span> <span m="3865860">is</span> <span m="3866000">not</span> <span m="3866220">doing</span> <span m="3866490">much.</span> <span m="3868350">As</span> <span m="3868600">n</span> <span m="3868820">gets</span> <span m="3869070">big,</span> <span m="3869890">the</span> <span m="3870010">square</span> <span m="3870370">root</span> <span m="3870490">of</span> <span m="3870610">n</span> <span m="3870730">grows,</span> <span m="3872060">but</span> <span m="3872270">it's</span> <span m="3872430">nothing</span> <span m="3872920">like</span> <span m="3873070">what's</span> <span m="3873240">happening</span> <span m="3873610">here.</span> </p>
<p><span m="3874370">If</span> <span m="3874520">you</span> <span m="3874670">had</span> <span m="3874840">to</span> <span m="3874910">describe</span> <span m="3875420">to</span> <span m="3875490">somebody</span> <span m="3875820">what's</span> <span m="3876050">going</span> <span m="3876270">on</span> <span m="3876400">in</span> <span m="3876480">this</span> <span m="3876660">bound,</span> <span m="3878230">would</span> <span m="3878400">you</span> <span m="3878480">start</span> <span m="3878810">here</span> <span m="3879510">or</span> <span m="3879680">here?</span> <span m="3880710">No.</span> <span m="3880990">You'd</span> <span m="3881170">start</span> <span m="3881450">here.</span> <span m="3881650">This</span> <span m="3881880">is</span> <span m="3881980">the</span> <span m="3882110">action,</span> <span m="3883040">and</span> <span m="3883240">there's</span> <span m="3883420">a</span> <span m="3883480">little</span> <span m="3883720">bit--</span> <span m="3883980">the</span> <span m="3884060">rest</span> <span m="3884330">is</span> <span m="3884430">just</span> <span m="3884750">in</span> <span m="3884830">the</span> <span m="3884890">slop,</span> <span m="3885380">in</span> <span m="3885500">the</span> <span m="3885630">air.</span> <span m="3886600">And</span> <span m="3886840">so</span> <span m="3886940">now</span> <span m="3887110">I've</span> <span m="3887190">used</span> <span m="3887470">judgement</span> <span m="3887860">to</span> <span m="3887920">say</span> <span m="3888810">that</span> <span m="3888920">delta</span> <span m="3890040">is</span> <span m="3890270">somewhere</span> <span m="3891080">in</span> <span m="3891200">this</span> <span m="3891390">stuff</span> <span m="3891810">here.</span> </p>
<p><span m="3893640">What's</span> <span m="3893820">really</span> <span m="3894200">happening,</span> <span m="3894620">and</span> <span m="3894930">the</span> <span m="3895000">nice</span> <span m="3895085">thing</span> <span m="3895170">is</span> <span m="3895310">they</span> <span m="3895430">match.</span> <span m="3895830">The</span> <span m="3895910">lower</span> <span m="3896160">bound</span> <span m="3897130">and</span> <span m="3897240">the</span> <span m="3897320">upper</span> <span m="3897490">bound</span> <span m="3897750">match</span> <span m="3898150">on</span> <span m="3898250">that</span> <span m="3898430">term.</span> <span m="3899770">So</span> <span m="3899870">what</span> <span m="3899990">I</span> <span m="3900060">do</span> <span m="3900280">is</span> <span m="3900400">I</span> <span m="3900500">write</span> <span m="3902190">it</span> <span m="3902430">equals</span> <span m="3903550">this</span> <span m="3903890">term</span> <span m="3904750">plus</span> <span m="3905160">something</span> <span m="3905970">that's</span> <span m="3906150">smaller</span> <span m="3908370">and,</span> <span m="3908860">in</span> <span m="3908960">particular,</span> <span m="3910390">it's</span> <span m="3910650">between</span> <span m="3910970">1/3</span> <span m="3911340">and</span> <span m="3911520">square</span> <span m="3911660">root</span> <span m="3911800">of</span> <span m="3911905">n</span> <span m="3912010">minus</span> <span m="3912290">2/3.</span> <span m="3914080">All</span> <span m="3914110">right.</span> </p>
<p><span m="3914780">So</span> <span m="3914810">what</span> <span m="3914900">I'm</span> <span m="3915010">trying</span> <span m="3915450">to</span> <span m="3915520">capture</span> <span m="3916040">here</span> <span m="3916890">is</span> <span m="3918100">just</span> <span m="3918340">the</span> <span m="3918430">guts</span> <span m="3919010">of</span> <span m="3919100">what's</span> <span m="3919360">happening</span> <span m="3919830">to</span> <span m="3919940">this</span> <span m="3920140">function</span> <span m="3920730">as</span> <span m="3920920">n</span> <span m="3921070">grows,</span> <span m="3922040">and</span> <span m="3922220">the</span> <span m="3922300">guts</span> <span m="3922630">of</span> <span m="3922730">it</span> <span m="3923300">is</span> <span m="3923520">this.</span> <span m="3924530">It's</span> <span m="3924730">not</span> <span m="3924940">exactly</span> <span m="3925480">equal</span> <span m="3925720">to</span> <span m="3925820">2/3</span> <span m="3926550">times</span> <span m="3926800">n</span> <span m="3926900">to</span> <span m="3926995">the</span> <span m="3927090">3/2,</span> <span m="3928210">but</span> <span m="3928370">it's</span> <span m="3928530">close</span> <span m="3929510">and,</span> <span m="3929660">in</span> <span m="3929760">fact,</span> <span m="3930270">if</span> <span m="3930350">I</span> <span m="3930440">take</span> <span m="3930670">the</span> <span m="3930740">limit</span> <span m="3931230">of</span> <span m="3931480">this</span> <span m="3931880">over</span> <span m="3932230">that,</span> <span m="3932585">that</span> <span m="3932940">limit</span> <span m="3933180">goes</span> <span m="3933380">to</span> <span m="3933470">1.</span> </p>
<p><span m="3934930">It's</span> <span m="3935110">a</span> <span m="3935160">way</span> <span m="3935460">of</span> <span m="3935550">saying</span> <span m="3936020">they're</span> <span m="3936790">approximately</span> <span m="3937670">the</span> <span m="3937770">same</span> <span m="3939060">that's</span> <span m="3939260">called</span> <span m="3939540">asymptotically</span> <span m="3940450">the</span> <span m="3940550">same,</span> <span m="3941760">and</span> <span m="3941890">we'll</span> <span m="3941980">talk</span> <span m="3942170">a</span> <span m="3942220">lot</span> <span m="3942560">more</span> <span m="3942780">about</span> <span m="3943000">asymptotic</span> <span m="3943620">notation</span> <span m="3944760">next</span> <span m="3944910">time.</span> <span m="3945150">We'll</span> <span m="3945360">give</span> <span m="3945540">you</span> <span m="3945610">five</span> <span m="3945970">more</span> <span m="3946140">symbols</span> <span m="3946560">besides</span> <span m="3947120">tilde</span> <span m="3948380">that</span> <span m="3948500">people</span> <span m="3948760">use.</span> <span m="3951900">Any</span> <span m="3952050">questions</span> <span m="3952540">about--</span> <span m="3953250">maybe</span> <span m="3953500">start</span> <span m="3953770">with</span> <span m="3953870">the</span> <span m="3953930">bounds.</span> <span m="3954670">Any</span> <span m="3954860">question</span> <span m="3955230">on</span> <span m="3955310">the</span> <span m="3955380">bounds</span> <span m="3955730">that</span> <span m="3955820">we</span> <span m="3955900">got?</span> <span m="3957110">That's</span> <span m="3957350">the</span> <span m="3957470">integration</span> <span m="3957970">method,</span> <span m="3958250">first</span> <span m="3958570">getting</span> <span m="3958760">the</span> <span m="3958830">bounds.</span> <span m="3961790">You</span> <span m="3961900">take</span> <span m="3962150">the</span> <span m="3962280">integral,</span> <span m="3964050">you</span> <span m="3964235">add</span> <span m="3964420">f</span> <span m="3964580">of</span> <span m="3964680">1</span> <span m="3964920">for</span> <span m="3964990">the</span> <span m="3965070">lower</span> <span m="3965320">bound,</span> <span m="3965650">you</span> <span m="3965730">add</span> <span m="3966010">f</span> <span m="3966190">of</span> <span m="3966335">n</span> <span m="3966480">for</span> <span m="3966550">the</span> <span m="3966660">upper</span> <span m="3966830">bound,</span> <span m="3968180">and</span> <span m="3968330">it's</span> <span m="3968480">in</span> <span m="3968580">between,</span> <span m="3969010">somewhere</span> <span m="3969360">in</span> <span m="3969440">there.</span> <span m="3972570">Questions</span> <span m="3972940">there?</span> </p>
<p><span m="3974960">All</span> <span m="3975160">right.</span> <span m="3975470">Then</span> <span m="3975670">we</span> <span m="3975770">plugged</span> <span m="3976100">it</span> <span m="3976170">in,</span> <span m="3977630">and</span> <span m="3977800">now</span> <span m="3978090">we</span> <span m="3978590">look</span> <span m="3978830">at</span> <span m="3979420">this</span> <span m="3979690">tilde</span> <span m="3980040">notation</span> <span m="3980590">that</span> <span m="3980720">says--</span> <span m="3981860">well,</span> <span m="3981960">first</span> <span m="3982210">we'd</span> <span m="3982300">write</span> <span m="3982470">it</span> <span m="3982540">like</span> <span m="3982700">this.</span> <span m="3982940">The</span> <span m="3983100">sum</span> <span m="3983510">is</span> <span m="3984120">this</span> <span m="3984340">value</span> <span m="3984650">plus</span> <span m="3984910">an</span> <span m="3985000">error</span> <span m="3985270">term</span> <span m="3986030">and,</span> <span m="3986240">lo</span> <span m="3986400">and</span> <span m="3986500">behold,</span> <span m="3987390">that</span> <span m="3987620">error</span> <span m="3987890">term</span> <span m="3988150">is</span> <span m="3988230">small.</span> <span m="3990840">If</span> <span m="3990875">I</span> <span m="3990910">take</span> <span m="3991150">the</span> <span m="3991220">limit</span> <span m="3991690">of</span> <span m="3991850">the</span> <span m="3991930">whole</span> <span m="3992180">thing</span> <span m="3992570">divided</span> <span m="3992960">by</span> <span m="3993250">the</span> <span m="3993670">big</span> <span m="3993920">term,</span> <span m="3994480">I</span> <span m="3994590">get</span> <span m="3994800">1,</span> <span m="3995760">which</span> <span m="3995940">means</span> <span m="3996170">this</span> <span m="3996360">thing</span> <span m="3996560">is</span> <span m="3997090">really</span> <span m="3997440">not</span> <span m="3997620">important.</span> <span m="3998780">So</span> <span m="3998940">I</span> <span m="3999000">write</span> <span m="3999250">this.</span> <span m="4002630">Questions</span> <span m="4003090">on</span> <span m="4003220">that?</span> <span m="4005840">All</span> <span m="4006090">right.</span> </p>
<p><span m="4008500">There's</span> <span m="4008710">one</span> <span m="4009050">more</span> <span m="4009480">case</span> <span m="4009780">to</span> <span m="4009880">consider,</span> <span m="4011560">and</span> <span m="4011750">now</span> <span m="4011830">we're</span> <span m="4011910">going</span> <span m="4011990">to</span> <span m="4012050">go</span> <span m="4012160">back</span> <span m="4012560">to</span> <span m="4012740">the</span> <span m="4012860">integration</span> <span m="4013380">bounds,</span> <span m="4013750">and</span> <span m="4013850">that</span> <span m="4014070">is</span> <span m="4014450">when</span> <span m="4014950">f</span> <span m="4015330">is</span> <span m="4015510">a</span> <span m="4015890">decreasing</span> <span m="4016520">function,</span> <span m="4017780">and</span> <span m="4017860">we're</span> <span m="4017940">going</span> <span m="4018120">to</span> <span m="4018190">do</span> <span m="4018350">the</span> <span m="4018450">analysis</span> <span m="4018930">for</span> <span m="4019050">that,</span> <span m="4019390">and</span> <span m="4019490">then</span> <span m="4019500">we'll</span> <span m="4019610">be</span> <span m="4019740">all</span> <span m="4019930">done.</span> </p>
<p><span m="4024480">So</span> <span m="4024550">we're</span> <span m="4024610">going</span> <span m="4024730">to</span> <span m="4024790">look</span> <span m="4025030">at</span> <span m="4025250">integration</span> <span m="4025910">bounds</span> <span m="4029740">when</span> <span m="4030140">f</span> <span m="4031170">is</span> <span m="4031650">decreasing</span> <span m="4032690">and</span> <span m="4032910">positive</span> <span m="4033370">still.</span> <span m="4045660">The</span> <span m="4045790">example</span> <span m="4046290">here</span> <span m="4046650">might</span> <span m="4046900">be,</span> <span m="4048190">for</span> <span m="4048220">example,</span> <span m="4053540">the</span> <span m="4053580">sum</span> <span m="4054450">i</span> <span m="4054810">equals</span> <span m="4055190">1</span> <span m="4055470">to</span> <span m="4055570">n,</span> <span m="4056750">1</span> <span m="4057250">over</span> <span m="4057640">the</span> <span m="4057730">square</span> <span m="4058020">root</span> <span m="4058140">of</span> <span m="4058270">i.</span> <span m="4060260">Say</span> <span m="4060590">you</span> <span m="4060720">had</span> <span m="4060970">to</span> <span m="4061040">get</span> <span m="4061310">some</span> <span m="4061660">idea</span> <span m="4062050">of</span> <span m="4062130">how</span> <span m="4062370">fast</span> <span m="4062830">that</span> <span m="4063020">function</span> <span m="4063430">is</span> <span m="4063550">growing</span> <span m="4064600">as</span> <span m="4064770">a</span> <span m="4064820">function</span> <span m="4065190">of</span> <span m="4065300">n.</span> <span m="4065970">I'm</span> <span m="4066190">summing</span> <span m="4066720">the</span> <span m="4066850">first</span> <span m="4068150">n</span> <span m="4070120">inverse</span> <span m="4070630">as</span> <span m="4070750">the</span> <span m="4070810">square</span> <span m="4071140">roots.</span> <span m="4073030">What</span> <span m="4073210">is</span> <span m="4073330">that</span> <span m="4073550">roughly</span> <span m="4073960">going</span> <span m="4074090">to</span> <span m="4074180">be?</span> <span m="4074370">How</span> <span m="4074530">fast</span> <span m="4074880">does</span> <span m="4074935">that</span> <span m="4074990">grow</span> <span m="4075185">as</span> <span m="4075380">a</span> <span m="4075420">function</span> <span m="4075780">of</span> <span m="4075880">n?</span> <span m="4077860">So</span> <span m="4078210">let's</span> <span m="4078380">do</span> <span m="4078550">that.</span> </p>
<p><span m="4082510">And,</span> <span m="4082640">of</span> <span m="4082700">course,</span> <span m="4083000">1</span> <span m="4083150">over</span> <span m="4083330">square</span> <span m="4083520">root</span> <span m="4083620">of</span> <span m="4083720">i</span> <span m="4083880">decreases</span> <span m="4085330">as</span> <span m="4085520">i</span> <span m="4085670">gets</span> <span m="4085890">bigger.</span> <span m="4088500">So</span> <span m="4088700">let's</span> <span m="4088960">do</span> <span m="4089090">the</span> <span m="4089200">general</span> <span m="4089677">picture</span> <span m="4090631">again</span> <span m="4094240">and</span> <span m="4094300">see</span> <span m="4094480">what</span> <span m="4094640">happens.</span> <span m="4095990">So</span> <span m="4096149">we</span> <span m="4096240">have</span> <span m="4096370">0,</span> <span m="4097840">1,</span> <span m="4098750">2,</span> <span m="4099790">3,</span> <span m="4101700">n</span> <span m="4101920">minus</span> <span m="4102270">2,</span> <span m="4103290">n</span> <span m="4103460">minus</span> <span m="4103810">1,</span> <span m="4104979">n,</span> <span m="4106939">and</span> <span m="4107120">now</span> <span m="4108220">f</span> <span m="4108479">of</span> <span m="4108634">n</span> <span m="4108790">is</span> <span m="4108890">the</span> <span m="4108970">small</span> <span m="4109340">term</span> <span m="4111550">and</span> <span m="4111990">f</span> <span m="4112189">of</span> <span m="4112330">n</span> <span m="4112450">minus</span> <span m="4112790">1,</span> <span m="4115450">f</span> <span m="4115830">of</span> <span m="4116010">3</span> <span m="4116689">here,</span> <span m="4121040">f</span> <span m="4121250">of</span> <span m="4121359">1.</span> </p>
<p><span m="4123090">Now,</span> <span m="4123910">I'm</span> <span m="4123979">going</span> <span m="4124100">to</span> <span m="4124160">draw</span> <span m="4124490">the</span> <span m="4124600">rectangle.</span> <span m="4125250">This</span> <span m="4125590">has</span> <span m="4125899">area</span> <span m="4126359">f</span> <span m="4126620">of</span> <span m="4126720">1.</span> <span m="4129470">This</span> <span m="4129750">one</span> <span m="4130080">has</span> <span m="4130470">area</span> <span m="4131399">f</span> <span m="4131609">of</span> <span m="4131700">2.</span> <span m="4135080">This</span> <span m="4135370">one</span> <span m="4135680">has</span> <span m="4135939">area</span> <span m="4136380">f</span> <span m="4136600">of</span> <span m="4136720">3,</span> <span m="4141210">and</span> <span m="4141550">then</span> <span m="4141760">this</span> <span m="4141990">one</span> <span m="4143130">has</span> <span m="4143569">area</span> <span m="4145229">f</span> <span m="4145410">of</span> <span m="4145529">n</span> <span m="4145660">minus</span> <span m="4146010">1</span> <span m="4146450">here,</span> <span m="4149640">and</span> <span m="4149870">then,</span> <span m="4150029">lastly,</span> <span m="4150710">area</span> <span m="4151050">f</span> <span m="4151310">of</span> <span m="4151430">n.</span> <span m="4154279">So</span> <span m="4154460">the</span> <span m="4154569">sum</span> <span m="4155689">is</span> <span m="4156029">the</span> <span m="4156180">area</span> <span m="4156620">in</span> <span m="4156819">the</span> <span m="4156910">rectangles,</span> <span m="4157500">just</span> <span m="4157770">like</span> <span m="4157939">before,</span> <span m="4158819">except</span> <span m="4159050">now</span> <span m="4159229">the</span> <span m="4159330">rectangles</span> <span m="4159790">are</span> <span m="4159840">getting</span> <span m="4160050">smaller.</span> <span m="4162020">Let's</span> <span m="4162260">draw</span> <span m="4162510">the</span> <span m="4162660">integral</span> <span m="4164569">like</span> <span m="4164800">we</span> <span m="4164880">did</span> <span m="4165029">before.</span> <span m="4167350">The</span> <span m="4168350">integral</span> <span m="4168870">is</span> <span m="4168950">the</span> <span m="4169080">area</span> <span m="4169510">under</span> <span m="4169760">this</span> <span m="4170000">curve,</span> <span m="4170660">f</span> <span m="4170870">of</span> <span m="4170979">x</span> <span m="4171290">here,</span> <span m="4172380">just</span> <span m="4172760">like</span> <span m="4172930">before.</span> <span m="4175960">So</span> <span m="4176120">this</span> <span m="4176319">is</span> <span m="4176450">f</span> <span m="4176620">of</span> <span m="4176729">x,</span> <span m="4178270">only</span> <span m="4178520">it's</span> <span m="4178660">decreasing.</span> </p>
<p><span m="4181840">Now,</span> <span m="4183200">let's</span> <span m="4183590">take</span> <span m="4183859">the</span> <span m="4184029">area</span> <span m="4184550">under</span> <span m="4184850">this</span> <span m="4185069">curve,</span> <span m="4188560">and</span> <span m="4188770">add</span> <span m="4189149">f</span> <span m="4189340">of</span> <span m="4189430">1</span> <span m="4189670">to</span> <span m="4189850">it.</span> <span m="4191540">If</span> <span m="4191710">I</span> <span m="4191859">take</span> <span m="4192130">the</span> <span m="4192279">area</span> <span m="4192670">under</span> <span m="4192899">this</span> <span m="4193354">curve,</span> <span m="4194460">all</span> <span m="4194720">the</span> <span m="4194780">way</span> <span m="4194910">down</span> <span m="4195150">to</span> <span m="4195220">here</span> <span m="4197360">and</span> <span m="4197510">then</span> <span m="4197700">add</span> <span m="4199040">f</span> <span m="4199280">of</span> <span m="4199390">1,</span> <span m="4200000">what</span> <span m="4200180">do</span> <span m="4200290">I</span> <span m="4200390">get?</span> <span m="4204440">Upper</span> <span m="4204760">bound</span> <span m="4205870">on</span> <span m="4206150">my</span> <span m="4206290">sum.</span> <span m="4207840">OK.</span> <span m="4208290">So</span> <span m="4208490">the</span> <span m="4208600">sum</span> <span m="4209910">i</span> <span m="4210170">equals</span> <span m="4210600">1</span> <span m="4210880">to</span> <span m="4211330">n</span> <span m="4211780">f</span> <span m="4212230">of</span> <span m="4212640">i</span> <span m="4213460">is</span> <span m="4213910">upper</span> <span m="4214220">bounded</span> <span m="4214720">by</span> <span m="4216670">that</span> <span m="4216940">guy,</span> <span m="4217270">which</span> <span m="4217370">is</span> <span m="4217460">f</span> <span m="4217650">of</span> <span m="4217750">1,</span> <span m="4219610">plus</span> <span m="4220100">my</span> <span m="4220270">integral,</span> <span m="4224850">which</span> <span m="4225010">is</span> <span m="4225120">the</span> <span m="4225260">area</span> <span m="4225540">under</span> <span m="4225680">the</span> <span m="4225760">curve.</span> <span m="4227520">The</span> <span m="4227690">integral</span> <span m="4228060">is</span> <span m="4228140">the</span> <span m="4228270">area</span> <span m="4228530">under</span> <span m="4228720">that</span> <span m="4228970">curve,</span> <span m="4229540">starting</span> <span m="4230020">here,</span> <span m="4231400">and</span> <span m="4231520">that</span> <span m="4232360">contains</span> <span m="4232990">all</span> <span m="4233230">these</span> <span m="4233480">rectangles,</span> <span m="4235110">and</span> <span m="4235390">then</span> <span m="4235510">I</span> <span m="4235640">just</span> <span m="4235860">add</span> <span m="4236090">in</span> <span m="4236500">f</span> <span m="4236710">of</span> <span m="4236800">1</span> <span m="4237180">to</span> <span m="4237290">get</span> <span m="4237420">an</span> <span m="4237480">upper</span> <span m="4237650">bound.</span> </p>
<p><span m="4241540">Now,</span> <span m="4241820">for</span> <span m="4241940">the</span> <span m="4242010">lower</span> <span m="4242340">bound,</span> <span m="4243430">think</span> <span m="4243710">about</span> <span m="4244060">shifting</span> <span m="4244750">the</span> <span m="4244840">whole</span> <span m="4245340">curve</span> <span m="4245910">left</span> <span m="4246440">by</span> <span m="4246630">one.</span> <span m="4247830">That</span> <span m="4248080">goes</span> <span m="4248280">to</span> <span m="4248370">there,</span> <span m="4249430">this</span> <span m="4249630">goes</span> <span m="4249880">to</span> <span m="4249970">here,</span> <span m="4251190">that</span> <span m="4251390">goes</span> <span m="4251660">to</span> <span m="4251750">here,</span> <span m="4252800">that</span> <span m="4253010">goes</span> <span m="4253260">to</span> <span m="4253340">there,</span> <span m="4253840">and</span> <span m="4254240">that</span> <span m="4254540">goes</span> <span m="4254770">to</span> <span m="4254860">there.</span> <span m="4256740">The</span> <span m="4257210">area</span> <span m="4257600">under</span> <span m="4257750">the</span> <span m="4257840">curve</span> <span m="4258210">did</span> <span m="4258340">not</span> <span m="4258740">change</span> <span m="4259200">when</span> <span m="4259320">I</span> <span m="4259410">shifted</span> <span m="4259870">it</span> <span m="4259980">left</span> <span m="4260240">by</span> <span m="4260390">one.</span> <span m="4262710">This</span> <span m="4262980">is</span> <span m="4263060">now</span> <span m="4263310">my</span> <span m="4263490">area</span> <span m="4263890">under</span> <span m="4264060">the</span> <span m="4264160">curve.</span> <span m="4266760">Stops</span> <span m="4267180">here.</span> <span m="4268830">What</span> <span m="4269060">do</span> <span m="4269170">I</span> <span m="4269260">get</span> <span m="4269480">when</span> <span m="4269600">I</span> <span m="4269660">take</span> <span m="4269970">that</span> <span m="4270260">area</span> <span m="4271310">and</span> <span m="4271500">add</span> <span m="4271690">in</span> <span m="4271820">this</span> <span m="4271980">last</span> <span m="4272250">box,</span> <span m="4272800">f</span> <span m="4273271">of</span> <span m="4273742">n?</span> <span m="4276570">A</span> <span m="4276690">lower</span> <span m="4277050">bound,</span> <span m="4278540">because</span> <span m="4278750">it's</span> <span m="4278920">contained</span> <span m="4279780">in</span> <span m="4279970">all</span> <span m="4280190">the</span> <span m="4280280">rectangles.</span> </p>
<p><span m="4290900">Now,</span> <span m="4291120">what's</span> <span m="4291410">really</span> <span m="4291840">weird</span> <span m="4293340">about</span> <span m="4293720">these</span> <span m="4293940">formulas,</span> <span m="4296520">do</span> <span m="4296640">they</span> <span m="4296780">look</span> <span m="4296930">familiar?</span> <span m="4300280">Yeah.</span> <span m="4301780">Yeah.</span> <span m="4303300">I</span> <span m="4303430">switched</span> <span m="4303760">them.</span> <span m="4305290">Yeah.</span> <span m="4305460">They're</span> <span m="4305560">the</span> <span m="4305660">same</span> <span m="4306010">formulas</span> <span m="4306470">we</span> <span m="4306610">had</span> <span m="4306810">over</span> <span m="4307020">here,</span> <span m="4308870">except</span> <span m="4309630">we</span> <span m="4309780">switched</span> <span m="4310090">the</span> <span m="4310160">direction</span> <span m="4310770">on</span> <span m="4310850">the</span> <span m="4310910">less</span> <span m="4311150">than</span> <span m="4311350">and</span> <span m="4311460">greater</span> <span m="4311730">than</span> <span m="4312040">signs.</span> <span m="4313970">Well,</span> <span m="4314130">I</span> <span m="4314190">swapped</span> <span m="4314670">f</span> <span m="4314830">of</span> <span m="4314920">1</span> <span m="4315130">and</span> <span m="4315220">f</span> <span m="4315380">of</span> <span m="4315540">n,</span> <span m="4315700">however</span> <span m="4315930">you</span> <span m="4316000">want</span> <span m="4316140">to</span> <span m="4316210">think</span> <span m="4316390">about</span> <span m="4316630">it.</span> <span m="4317280">The</span> <span m="4318070">lower</span> <span m="4318500">bound</span> <span m="4319020">here</span> <span m="4320010">in</span> <span m="4320100">that</span> <span m="4320280">case</span> <span m="4320570">became</span> <span m="4320910">the</span> <span m="4321040">upper</span> <span m="4321230">bound</span> <span m="4322250">in</span> <span m="4322390">this</span> <span m="4322560">case.</span> <span m="4325000">Is</span> <span m="4325070">that</span> <span m="4325290">possible</span> <span m="4325780">that</span> <span m="4326250">the</span> <span m="4326360">lower</span> <span m="4326570">bound</span> <span m="4326780">became</span> <span m="4327050">the</span> <span m="4327140">upper</span> <span m="4327300">bound?</span> <span m="4329330">Yeah.</span> <span m="4329880">Yeah,</span> <span m="4330650">because</span> <span m="4330970">what</span> <span m="4331170">really</span> <span m="4331470">happened</span> <span m="4331820">here--</span> <span m="4331910">which</span> <span m="4332090">is</span> <span m="4332190">the</span> <span m="4332270">big</span> <span m="4332510">term</span> <span m="4332790">in</span> <span m="4332860">this</span> <span m="4333020">case,</span> <span m="4333300">fn</span> <span m="4333660">or</span> <span m="4333760">f1?</span> <span m="4335240">f1</span> <span m="4335650">is</span> <span m="4335720">the</span> <span m="4335790">big</span> <span m="4336000">term</span> <span m="4336280">because</span> <span m="4336440">it's</span> <span m="4336580">decreasing,</span> <span m="4337290">so</span> <span m="4337390">it's</span> <span m="4337590">totally</span> <span m="4338180">symmetric.</span> <span m="4339650">All</span> <span m="4339830">right?</span> </p>
<p><span m="4340680">The</span> <span m="4340810">proof</span> <span m="4341160">was</span> <span m="4341300">very</span> <span m="4341560">similar,</span> <span m="4342370">so</span> <span m="4342510">the</span> <span m="4342600">nice</span> <span m="4342830">thing</span> <span m="4342970">is</span> <span m="4343090">you've</span> <span m="4343290">only</span> <span m="4343345">got</span> <span m="4343400">to</span> <span m="4343460">remember</span> <span m="4344400">the</span> <span m="4344510">bounds</span> <span m="4344850">are</span> <span m="4344900">now</span> <span m="4345120">simple</span> <span m="4345710">for</span> <span m="4345870">any</span> <span m="4346060">sum</span> <span m="4346440">as</span> <span m="4346550">long</span> <span m="4346720">as</span> <span m="4346840">an</span> <span m="4346970">increasing</span> <span m="4347490">or</span> <span m="4347530">decreasing,</span> <span m="4348550">it's</span> <span m="4348730">the</span> <span m="4348820">same</span> <span m="4349120">as</span> <span m="4349230">the</span> <span m="4349360">integral.</span> <span m="4350490">The</span> <span m="4350620">lower</span> <span m="4350920">bound</span> <span m="4351260">is</span> <span m="4351360">the</span> <span m="4351440">smaller</span> <span m="4352860">of</span> <span m="4353010">the</span> <span m="4353070">first</span> <span m="4353370">and</span> <span m="4353460">last</span> <span m="4353670">term,</span> <span m="4353920">and</span> <span m="4353965">the</span> <span m="4354010">upper</span> <span m="4354150">bounds</span> <span m="4354420">are</span> <span m="4354480">larger</span> <span m="4354900">of</span> <span m="4354970">the</span> <span m="4355040">first</span> <span m="4355330">and</span> <span m="4355420">last</span> <span m="4355680">term.</span> <span m="4356650">Very</span> <span m="4357070">easy</span> <span m="4357400">to</span> <span m="4357490">remember.</span> <span m="4357930">Probably</span> <span m="4358180">don't</span> <span m="4358260">even</span> <span m="4358340">need</span> <span m="4358550">the</span> <span m="4358610">crib</span> <span m="4358810">sheet</span> <span m="4359040">for</span> <span m="4359240">it,</span> <span m="4359770">although</span> <span m="4360080">to</span> <span m="4360170">be</span> <span m="4360260">safe,</span> <span m="4361481">want</span> <span m="4361890">to</span> <span m="4362368">write</span> <span m="4362846">that</span> <span m="4363324">down.</span> <span m="4366200">So</span> <span m="4366310">now</span> <span m="4366650">it's</span> <span m="4366780">easy</span> <span m="4367050">to</span> <span m="4367120">compute</span> <span m="4368520">good</span> <span m="4368810">bounds</span> <span m="4369340">on</span> <span m="4369520">the</span> <span m="4369600">sum</span> <span m="4370060">of</span> <span m="4370340">the</span> <span m="4370560">inverse</span> <span m="4371055">square</span> <span m="4371302">roots.</span> <span m="4382020">Any</span> <span m="4382190">questions</span> <span m="4383390">there</span> <span m="4383570">before</span> <span m="4383880">I</span> <span m="4383980">go</span> <span m="4386450">do</span> <span m="4386650">it?</span> </p>
<p><span m="4388530">So</span> <span m="4388880">let's</span> <span m="4389100">take</span> <span m="4389270">the</span> <span m="4389340">case</span> <span m="4389770">where</span> <span m="4390000">we're</span> <span m="4390120">summing</span> <span m="4391170">1</span> <span m="4391440">over</span> <span m="4391660">square</span> <span m="4391900">root</span> <span m="4392010">of</span> <span m="4392140">i.</span> <span m="4395360">So</span> <span m="4395820">we</span> <span m="4395910">compute</span> <span m="4396240">the</span> <span m="4396430">integral</span> <span m="4398640">of</span> <span m="4398840">1</span> <span m="4399050">over</span> <span m="4399220">square</span> <span m="4399450">root</span> <span m="4399570">of</span> <span m="4399660">x</span> <span m="4400630">dx.</span> <span m="4402970">That</span> <span m="4403220">equals</span> <span m="4403810">the</span> <span m="4403890">square</span> <span m="4404130">root</span> <span m="4404250">of</span> <span m="4404360">x</span> <span m="4404650">over</span> <span m="4404940">1/2,</span> <span m="4406530">evaluated</span> <span m="4407130">at</span> <span m="4407250">n</span> <span m="4407445">and</span> <span m="4407640">1.</span> <span m="4409500">That</span> <span m="4409740">equals</span> <span m="4410610">2</span> <span m="4411210">square</span> <span m="4411600">root</span> <span m="4411665">of</span> <span m="4411730">n</span> <span m="4412140">minus</span> <span m="4412520">1,</span> <span m="4413650">or</span> <span m="4413840">two</span> <span m="4414160">square</span> <span m="4414280">root</span> <span m="4414400">of</span> <span m="4414520">n</span> <span m="4415930">minus</span> <span m="4416190">2,</span> <span m="4418300">and</span> <span m="4418620">now</span> <span m="4418910">we</span> <span m="4419040">can</span> <span m="4419510">bound</span> <span m="4420030">the</span> <span m="4420120">sum.</span> <span m="4428650">The</span> <span m="4428780">upper</span> <span m="4428960">bound</span> <span m="4429440">is</span> <span m="4430020">f</span> <span m="4430240">of</span> <span m="4430350">1</span> <span m="4430960">plus</span> <span m="4432280">the</span> <span m="4432420">integral.</span> <span m="4434380">The</span> <span m="4434480">lower</span> <span m="4434780">bound</span> <span m="4435940">is</span> <span m="4436570">f</span> <span m="4436780">of</span> <span m="4436820">n</span> <span m="4438140">plus</span> <span m="4438460">the</span> <span m="4438550">integral.</span> <span m="4442420">What</span> <span m="4442670">is</span> <span m="4443010">f</span> <span m="4443163">of</span> <span m="4443316">1?</span> <span m="4445770">One?</span> <span m="4448070">One</span> <span m="4448140">over</span> <span m="4448183">the</span> <span m="4448226">square</span> <span m="4448270">root</span> <span m="4448490">of</span> <span m="4448570">1</span> <span m="4448770">is</span> <span m="4448890">1.</span> <span m="4449530">What</span> <span m="4449760">is</span> <span m="4449980">f</span> <span m="4450133">of</span> <span m="4450286">n?</span> <span m="4454120">One</span> <span m="4454380">over</span> <span m="4454550">the</span> <span m="4454630">square</span> <span m="4454870">root</span> <span m="4454980">of</span> <span m="4455100">n.</span> <span m="4455400">Small.</span> </p>
<p><span m="4457350">So</span> <span m="4457480">these</span> <span m="4457750">bounds</span> <span m="4458110">are</span> <span m="4458170">pretty</span> <span m="4458550">close</span> <span m="4458990">here,</span> <span m="4459250">right?</span> <span m="4460290">In</span> <span m="4460400">fact,</span> <span m="4460730">this</span> <span m="4460950">gets</span> <span m="4461160">really</span> <span m="4461470">tiny</span> <span m="4461820">as</span> <span m="4461910">n</span> <span m="4462080">gets</span> <span m="4462280">big,</span> <span m="4462570">so</span> <span m="4462700">I'm</span> <span m="4462790">just</span> <span m="4462930">going</span> <span m="4463050">to</span> <span m="4463110">replace</span> <span m="4463520">this</span> <span m="4463730">with</span> <span m="4464120">2</span> <span m="4464330">square</span> <span m="4464480">root</span> <span m="4464630">of</span> <span m="4464785">n</span> <span m="4464940">minus</span> <span m="4465290">2</span> <span m="4465590">and</span> <span m="4465690">make</span> <span m="4465840">it</span> <span m="4465910">a</span> <span m="4465940">strict</span> <span m="4466320">lower</span> <span m="4466570">bound,</span> <span m="4467880">and</span> <span m="4468100">this--</span> <span m="4469450">cancel</span> <span m="4469870">there--</span> <span m="4470020">I</span> <span m="4470185">get</span> <span m="4470350">2</span> <span m="4470590">square</span> <span m="4470970">root</span> <span m="4471015">of</span> <span m="4471060">n</span> <span m="4472110">minus</span> <span m="4472500">1.</span> <span m="4474350">Wow,</span> <span m="4474740">these</span> <span m="4475140">bounds</span> <span m="4475430">are</span> <span m="4475480">great.</span> <span m="4476050">They're</span> <span m="4476190">within</span> <span m="4476430">one</span> <span m="4478040">for</span> <span m="4478270">all</span> <span m="4478540">n.</span> <span m="4480280">That's</span> <span m="4480580">really</span> <span m="4480810">good.</span> </p>
<p><span m="4483260">So</span> <span m="4484240">we</span> <span m="4484500">can</span> <span m="4484710">rewrite</span> <span m="4485220">this</span> <span m="4485650">in</span> <span m="4485750">terms</span> <span m="4486080">of</span> <span m="4487430">what</span> <span m="4487840">really</span> <span m="4488080">matters.</span> <span m="4488570">What</span> <span m="4489030">really</span> <span m="4489290">matters</span> <span m="4489680">in</span> <span m="4489750">these</span> <span m="4489930">bounds?</span> <span m="4492510">How</span> <span m="4492750">fast</span> <span m="4493140">is</span> <span m="4493240">this</span> <span m="4493410">function</span> <span m="4493780">growing?</span> </p>
<p><span m="4494560">AUDIENCE:</span> <span m="4494880">2</span> <span m="4494986">root</span> <span m="4495093">n.</span> </p>
<p><span m="4495420">PROFESSOR: 2</span> <span m="4495570">root</span> <span m="4495770">n.</span> <span m="4496000">That's</span> <span m="4496220">what</span> <span m="4496330">really</span> <span m="4496560">matters,</span> <span m="4496990">so</span> <span m="4497165">let's</span> <span m="4497340">write</span> <span m="4497503">that</span> <span m="4497666">down.</span> <span m="4506020">So</span> <span m="4506310">this</span> <span m="4506490">says</span> <span m="4506960">that</span> <span m="4507360">the</span> <span m="4507760">sum</span> <span m="4508310">i</span> <span m="4508620">equals</span> <span m="4508760">1</span> <span m="4509133">to</span> <span m="4509319">n</span> <span m="4509506">of</span> <span m="4509693">1</span> <span m="4509880">over</span> <span m="4510140">square</span> <span m="4510470">root</span> <span m="4510555">i</span> <span m="4513470">equals</span> <span m="4514780">2</span> <span m="4515180">square</span> <span m="4515530">root</span> <span m="4515585">of</span> <span m="4515640">n--</span> <span m="4516692">we</span> <span m="4516926">have</span> <span m="4517160">a</span> <span m="4517400">minus</span> <span m="4517880">delta</span> <span m="4518510">n,</span> <span m="4519770">where</span> <span m="4520440">delta</span> <span m="4521090">is</span> <span m="4521290">between</span> <span m="4523660">1</span> <span m="4524170">and</span> <span m="4524340">2.</span> <span m="4526990">And</span> <span m="4527170">so</span> <span m="4527460">if</span> <span m="4527570">I</span> <span m="4527640">use</span> <span m="4527930">the</span> <span m="4528010">tilde</span> <span m="4528370">notation,</span> <span m="4533600">what</span> <span m="4533820">would</span> <span m="4533920">I</span> <span m="4533990">write</span> <span m="4534250">down</span> <span m="4534520">here</span> <span m="4534690">for</span> <span m="4534810">the</span> <span m="4534880">tilde?</span> <span m="4536210">Past</span> <span m="4536520">the</span> <span m="4536770">tilde?</span> <span m="4539180">I</span> <span m="4539280">don't</span> <span m="4539400">want</span> <span m="4539530">to</span> <span m="4539590">mess--</span> </p>
<p><span m="4539970">AUDIENCE: Tilde.</span> </p>
<p><span m="4540110">PROFESSOR:  --I</span> <span m="4540410">don't</span> <span m="4540640">want</span> <span m="4540705">to</span> <span m="4540770">keep</span> <span m="4540806">track</span> <span m="4540843">of</span> <span m="4540880">all</span> <span m="4541026">the</span> <span m="4541173">delta</span> <span m="4541320">stuff</span> <span m="4541670">as</span> <span m="4541770">n</span> <span m="4541940">gets</span> <span m="4542070">big.</span> </p>
<p><span m="4542640">AUDIENCE:  2</span> <span m="4542900">root</span> <span m="4542980">n.</span> </p>
<p><span m="4543390">PROFESSOR: 2</span> <span m="4543520">root</span> <span m="4543645">n,</span> <span m="4545720">because</span> <span m="4547320">this</span> <span m="4547610">term</span> <span m="4548420">over</span> <span m="4548680">that</span> <span m="4549390">goes</span> <span m="4549650">to</span> <span m="4549750">0</span> <span m="4550260">as</span> <span m="4550390">n</span> <span m="4550880">gets</span> <span m="4551130">large,</span> <span m="4552010">so</span> <span m="4552160">let's</span> <span m="4552310">just</span> <span m="4552550">check</span> <span m="4552820">that.</span> <span m="4556630">So</span> <span m="4556790">we</span> <span m="4556880">take</span> <span m="4557170">the</span> <span m="4557260">limit</span> <span m="4557980">as</span> <span m="4558200">n</span> <span m="4558370">goes</span> <span m="4558620">to</span> <span m="4558740">infinity</span> <span m="4560360">of</span> <span m="4560840">2</span> <span m="4561240">root</span> <span m="4561530">n</span> <span m="4562300">minus</span> <span m="4562670">delta</span> <span m="4563030">n</span> <span m="4564600">over</span> <span m="4565000">2</span> <span m="4565260">root</span> <span m="4565470">n.</span> <span m="4566235">I'm</span> <span m="4566610">just</span> <span m="4566880">checking</span> <span m="4567190">the</span> <span m="4567270">definition</span> <span m="4567810">now.</span> <span m="4568150">That's</span> <span m="4568310">what</span> <span m="4568400">the</span> <span m="4568470">definition</span> <span m="4568950">would</span> <span m="4569060">be.</span> <span m="4571020">Equals</span> <span m="4571860">1</span> <span m="4573990">minus</span> <span m="4574550">the</span> <span m="4574640">limit</span> <span m="4576800">as</span> <span m="4576980">n</span> <span m="4577150">goes</span> <span m="4577380">to</span> <span m="4577490">infinity</span> <span m="4578410">of</span> <span m="4578620">2</span> <span m="4579190">over</span> <span m="4579520">2</span> <span m="4579800">root</span> <span m="4580030">n.</span> <span m="4581200">This</span> <span m="4581590">is</span> <span m="4581760">0.</span> <span m="4583250">So</span> <span m="4583450">it</span> <span m="4583520">equals</span> <span m="4583840">1.</span> <span m="4585750">And</span> <span m="4585880">so</span> <span m="4585980">now</span> <span m="4586190">you</span> <span m="4586340">know</span> <span m="4586770">that</span> <span m="4587970">the</span> <span m="4588140">sum</span> <span m="4588450">of</span> <span m="4588510">the</span> <span m="4588570">first</span> <span m="4588940">n</span> <span m="4589410">inverse</span> <span m="4589740">square</span> <span m="4589980">roots</span> <span m="4590390">grows</span> <span m="4591110">as</span> <span m="4591340">2</span> <span m="4591510">root</span> <span m="4591680">n,</span> <span m="4592120">which</span> <span m="4592310">is</span> <span m="4592430">the</span> <span m="4592520">integral.</span> <span m="4592900">Yeah.</span> </p>
<p><span m="4593357">AUDIENCE:</span> <span m="4593814">[INAUDIBLE]</span> <span m="4594271">dropped</span> <span m="4594500">off</span> <span m="4594730">the</span> <span m="4594870">lower</span> <span m="4595233">bound</span> <span m="4595960">that</span> <span m="4596316">f</span> <span m="4596434">of</span> <span m="4596553">n</span> <span m="4597030">was</span> <span m="4597210">1</span> <span m="4597365">over</span> <span m="4597520">root</span> <span m="4597676">n?</span> </p>
<p><span m="4598610">PROFESSOR: Yeah,</span> <span m="4599160">I</span> <span m="4599330">dropped</span> <span m="4599820">it</span> <span m="4599900">off,</span> <span m="4600720">because</span> <span m="4600900">it</span> <span m="4601030">was</span> <span m="4601670">so</span> <span m="4601970">tiny</span> <span m="4602400">and</span> <span m="4602520">going</span> <span m="4602640">to</span> <span m="4602710">zero,</span> <span m="4603140">I</span> <span m="4603210">just</span> <span m="4603470">made</span> <span m="4603623">a</span> <span m="4603776">strict</span> <span m="4603930">less</span> <span m="4604270">than.</span> <span m="4605150">In</span> <span m="4605250">fact,</span> <span m="4605590">yes.</span> <span m="4605850">I</span> <span m="4605940">don't</span> <span m="4606170">hurt</span> <span m="4606490">myself</span> <span m="4606950">by</span> <span m="4607990">dropping</span> <span m="4608480">it</span> <span m="4608580">off.</span> <span m="4608810">In</span> <span m="4608880">fact,</span> <span m="4609100">the</span> <span m="4609170">lower</span> <span m="4609420">bound</span> <span m="4609680">was</span> <span m="4609780">a</span> <span m="4609840">little</span> <span m="4610100">bit--</span> <span m="4611040">I</span> <span m="4611170">made</span> <span m="4611350">a</span> <span m="4611400">little</span> <span m="4611680">weaker</span> <span m="4612020">lower</span> <span m="4612260">bound.</span> <span m="4613210">So</span> <span m="4613360">this</span> <span m="4613560">is</span> <span m="4613640">still</span> <span m="4613880">true.</span> <span m="4615110">I</span> <span m="4615220">just--</span> <span m="4615460">it</span> <span m="4615570">wasn't</span> <span m="4615830">as</span> <span m="4615920">tight</span> <span m="4616130">as</span> <span m="4616230">it</span> <span m="4616320">used</span> <span m="4616520">to</span> <span m="4616580">be,</span> <span m="4616710">so</span> <span m="4616860">I</span> <span m="4616920">could</span> <span m="4617250">keep</span> <span m="4618290">it</span> <span m="4618360">around.</span> </p>
<p><span m="4619980">Yeah,</span> <span m="4620240">it</span> <span m="4620285">doesn't</span> <span m="4620330">hurt</span> <span m="4620470">to</span> <span m="4620550">keep</span> <span m="4620750">it</span> <span m="4620820">around,</span> <span m="4621140">then</span> <span m="4621250">it's</span> <span m="4621380">a</span> <span m="4621460">less</span> <span m="4621630">than</span> <span m="4621770">or</span> <span m="4621860">equal</span> <span m="4622120">there.</span> <span m="4624280">And</span> <span m="4624440">now</span> <span m="4624710">this</span> <span m="4624920">would</span> <span m="4625070">be</span> <span m="4630780">something</span> <span m="4631020">like</span> <span m="4631170">that.</span> <span m="4631430">So</span> <span m="4631520">I</span> <span m="4631560">could</span> <span m="4631770">keep</span> <span m="4631950">it</span> <span m="4632020">around,</span> <span m="4632240">but</span> <span m="4632330">I'm</span> <span m="4632390">going</span> <span m="4632510">to</span> <span m="4632570">get</span> <span m="4632730">rid</span> <span m="4632980">of</span> <span m="4633050">it</span> <span m="4633140">anyway,</span> <span m="4633430">because</span> <span m="4633580">I'm</span> <span m="4633610">going</span> <span m="4633730">to</span> <span m="4633790">go</span> <span m="4633900">to</span> <span m="4633970">the</span> <span m="4634060">tilde</span> <span m="4634350">notation,</span> <span m="4636430">and</span> <span m="4636830">as</span> <span m="4637210">n</span> <span m="4637370">gets</span> <span m="4637550">big,</span> <span m="4637800">this</span> <span m="4638000">is</span> <span m="4638110">really</span> <span m="4638370">tiny.</span> <span m="4639880">So</span> <span m="4639980">in</span> <span m="4640040">this</span> <span m="4640210">case,</span> <span m="4640450">the</span> <span m="4640520">bounds</span> <span m="4640870">are</span> <span m="4640910">great.</span> <span m="4641460">You</span> <span m="4641570">can</span> <span m="4641850">nail</span> <span m="4642230">it</span> <span m="4642340">pretty</span> <span m="4642580">much</span> <span m="4643410">right</span> <span m="4643700">on.</span> <span m="4646540">Yeah.</span> </p>
<p><span m="4647290">AUDIENCE:</span> <span m="4647790">You</span> <span m="4648290">said</span> <span m="4648456">one over</span> <span m="4648623">n</span> <span m="4648790">still there,</span> <span m="4650290">the</span> <span m="4650540">number</span> <span m="4650790">is</span> <span m="4650956">bigger</span> <span m="4651123">than</span> <span m="4651178">it</span> <span m="4651234">would</span> <span m="4651290">normally</span> <span m="4651790">be,</span> <span m="4652290">so</span> <span m="4652515">when</span> <span m="4652740">you</span> <span m="4653190">take</span> <span m="4653640">it</span> <span m="4653752">out,</span> <span m="4653865">it</span> <span m="4653977">becomes</span> <span m="4654090">smaller,</span> <span m="4654540">so</span> <span m="4654990">how</span> <span m="4655890">could</span> <span m="4656340">you</span> <span m="4656562">go</span> <span m="4656785">to</span> <span m="4656896">a</span> <span m="4657007">less</span> <span m="4657118">than</span> <span m="4657230">[INAUDIBLE]?</span> </p>
<p><span m="4659460">PROFESSOR: Well,</span> <span m="4659620">you're</span> <span m="4659730">saying</span> <span m="4659960">you</span> <span m="4660030">don't</span> <span m="4660230">like</span> <span m="4660510">the</span> <span m="4660580">fact</span> <span m="4660800">I</span> <span m="4660920">dropped</span> <span m="4661210">it</span> <span m="4661280">here?</span> </p>
<p><span m="4662246">AUDIENCE: Yes.</span> <span m="4662729">When</span> <span m="4662970">you</span> <span m="4663212">drop</span> <span m="4663453">it,</span> <span m="4663695">why</span> <span m="4663791">do</span> <span m="4663888">you</span> <span m="4663984">go</span> <span m="4664081">to</span> <span m="4664178">a</span> <span m="4664339">less</span> <span m="4664500">than</span> <span m="4664661">instead</span> <span m="4664902">of</span> <span m="4665144">[INAUDIBLE]?</span> </p>
<p><span m="4665627">PROFESSOR: Oh,</span> <span m="4666110">because</span> <span m="4666370">I've</span> <span m="4667140">got</span> <span m="4667330">a</span> <span m="4669220">bigger</span> <span m="4669460">bound</span> <span m="4669890">that</span> <span m="4670010">I</span> <span m="4670080">made</span> <span m="4670340">less</span> <span m="4671610">when</span> <span m="4671770">I</span> <span m="4671850">dropped</span> <span m="4672250">it.</span> <span m="4672480">I</span> <span m="4672600">took</span> <span m="4672830">something</span> <span m="4673110">away,</span> <span m="4673770">so</span> <span m="4673950">I</span> <span m="4674050">know</span> <span m="4674350">I</span> <span m="4674430">could</span> <span m="4674610">never</span> <span m="4674860">equal</span> <span m="4675300">this,</span> <span m="4675550">because</span> <span m="4675710">I</span> <span m="4675790">know</span> <span m="4675950">it's</span> <span m="4676090">bigger</span> <span m="4676350">than</span> <span m="4676500">this.</span> <span m="4678760">I</span> <span m="4678900">know</span> <span m="4679090">that</span> <span m="4679210">the</span> <span m="4679310">real</span> <span m="4679560">answer</span> <span m="4679820">has</span> <span m="4680030">to</span> <span m="4680090">be</span> <span m="4680170">at</span> <span m="4680260">least</span> <span m="4680540">this</span> <span m="4680740">big,</span> <span m="4681710">and</span> <span m="4681880">so</span> <span m="4682100">it</span> <span m="4682210">has</span> <span m="4682420">to</span> <span m="4682500">be</span> <span m="4682610">bigger</span> <span m="4683130">than</span> <span m="4683300">something</span> <span m="4683600">smaller.</span> <span m="4684990">That's</span> <span m="4685140">why</span> <span m="4685320">I</span> <span m="4685470">did</span> <span m="4685620">it.</span> <span m="4689960">Any</span> <span m="4690190">other</span> <span m="4690360">questions?</span> <span m="4693240">We'll</span> <span m="4693380">get</span> <span m="4693550">more</span> <span m="4693720">practice</span> <span m="4694220">tomorrow</span> <span m="4694740">and</span> <span m="4694880">next</span> <span m="4695180">time</span> <span m="4695280">with</span> <span m="4695380">this</span> <span m="4695480">stuff.</span> </p>
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